Higher Spherical Polynomials T. Ibukiyama and D. Zagier

Contents

Introduction 1

Chapter I. General Theory 5 §1. The generalized Legendre differential equation 5 §2. Dimension formula and decomposition theorem 8 §3. Scalar product and orthogonality 13 §4. The Lie algebra of mixed Laplacians 19 §5. The two canonical bases of P_{a}(d) 24 §6. Higher spherical polynomials and invariant harmonic polynomials 30

Chapter II. Explicit Constructions 35 §7. Construction of invariant harmonic polynomials for d = 4 35 §8. Higher spherical polynomials for n = 3 38

A. Generating function 38 B. Ratios of the various basis elements 41 C. Recursion relations 43 D. Explicit formulas for the coefficients of P_{ν ν ν,d} 45 E. Inhomogeneous coordinates and angular coordinates 48

§9. A universal generating function for the descending basis 50 A. Existence of the power series G^{(n)} 51 B. Proofs of the closed formula and commutation relation for L_{p} 56 C. Special cases and explicit formulas 59

§10. Algebraic generating functions for the case n = 4 67 A. Even values of d: experimental results 67 B. Algebraicity and non-algebraicity proofs for G^{(4)} when s is integral 69 C. Properties of the generating function for fixed integral values of s 74 D. Generating function with respect to s 78 E. Odd values of d 80 F. Final remarks 82

Tables of higher spherical polynomials 83 Table 1. Spherical polynomials for n = 3 83 Table 2. Spherical polynomials and Gram matrices for n = 4 and n = 5 85 Table 3. Determinants of Gram matrices 92

List of Notations 95

Bibliography 96 HIGHER SPHERICAL POLYNOMIALS

Tomoyoshi Ibukiyama and Don Zagier

Abstract. In this paper, we study a new class of special functions. Specifically, we study a vector space P^{(n)}(d) (n ∈ ℕ, d ∈ ℂ) of polynomials P (T) in n × n symmetric matrices T = (t_{ij}). For integral d ≥ n these are the polynomials on (ℝ^{d})^{n} that are invariant with respect to the diagonal action of O(d) and harmonic on each ℝ^{d}, and for general d they are the polynomial solutions of a certain system of differential equations. For n = 2 these are the classical Legendre and Gegenbauer polynomials, but for n > 2 they are new and are interesting for several reasons, including an application to the theory of Siegel modular forms and the fact that for n = 3 the associated system of differential equations is holonomic. When n = 3, each homogeneous component of P^{(n)}(d) is one-dimensional, so there is a canonical basis. Even here the structure of the polynomials turns out to be very subtle. For n > 3 there is no obvious basis. We construct two canonical bases, dual to one another, and a generating function (generalizing the classical one for n = 2, and algebraic if n = 3 but not in general) for one of them. We also provide tables that illustrate some of the idiosyncracies of the theory.

Introduction

For two natural numbers n and d and a scalar product ( , ) on ℝ^{d}, we have a map β_{n} from (ℝ^{d})^{n} to the space S_{n} of real symmetric n × n matrices, given by

β_{n} : (x_{1}, …, x_{n}) ↦ T = (t_{ij})_{1≤i,j≤n} , t_{ij} = (x_{i}, x_{j}) .  (1)

Thus any polynomial P on S_{n} defines a polynomial P̃ = P ∘ β_{n} on (ℝ^{d})^{n}. If d ≥ n, then we denote by P^{(n)}(d) the space of polynomials P on S_{n} for which P̃ is harmonic with respect to x_{i} for each i = 1, …, n. For each multidegree a = (a_{1}, …, a_{n}) ∈ (ℤ_{≥0})^{n} we denote by P_{a}(d) the subspace of P ∈ P^{(n)}(d) for which P̃ is a homogeneous of degree a_{i} with respect to x_{i} for each i = 1, …, n. (Here the superscript “(n)” can be omitted since n is just the length of a.) Later in the paper we will generalize the definitions of both P^{(n)}(d) and P_{a}(d) to arbitrary complex values of d. The elements of P^{(n)}(d) will be called higher spherical polynomials. There are several motivations for studying them:

(i) If n = 2, then P_{(a_{1},a_{2})}(d) is one-dimensional if a_{1} = a_{2} (it is 0 otherwise) and is then spanned by a polynomial in t_{11}t_{22} and t_{12} which is the homogeneous version of one of the classical families of orthogonal polynomials (Legendre polynomials if d = 3, Chebyshev polynomials if d = 2 or d = 4, and Gegenbauer polynomial for d arbitrary). These classical polynomials occur in many places in mathematics and mathematical physics and have many nice properties: differential equations, orthogonality properties, recursions, generating functions, closed formulas, etc. It is natural to try to generalize these.

(ii) The polynomials in P^{(n)}(d) have an application in the theory of Siegel modular forms. This is described in detail in [7] (similar differential operators are given in [2] and [4]) and will not be pursued in this paper, but was our original reason for studying this particular space of polynomials. Briefly, the connection is as follows. If P ∈ P_{a}(d) (d ≥ n) and if

F (Z) = ∑_{T} c(T) e^{2πi Tr(T Z)} (Z ∈ ℌ_{n} = Siegel upper half-space)

is a holomorphic Siegel modular form of weight d/2, then the function

P (\frac{1 + δ_{ij}}{2} \frac{∂}{∂z_{ij}} )F (Z)|_{ℌ_{1}^{n}} = ∑_{T} c(T) P (T) e^{2πi(t_{11}z_{1}+...+t_{nn}z_{n})}  (z_{1}, …, z_{n} ∈ ℌ_{1}) (2)

is an elliptic modular form of weight d/2 + a_{i} in z_{i} for each i = 1, …, n. (By considering images g(ℌ^{n}) ⊂ ℌ_{n}, where g is a suitable element of Sp_{2n}(ℝ), one also gets maps from Siegel modular forms of degree n to Hilbert modular forms on totally real number fields of degree n, or to products of Hilbert modular forms on several totally real fields with degrees adding up to n.) We can understand why this should be true by taking F to be a Siegel theta series, say

F (Z) = ∑_{(x_{1},... ,x_{n})∈L^{n}} e^{2πi Tr( β_{n}(x_{1},... ,x_{n}) Z)}

where L ⊂ ℝ^{d} is a lattice on which the quadratic form (x, x) takes rational values. Then the function defined by (2) is the series

∑_{x_{1}∈L} ⋯ ∑_{x_{n}∈L} P̃(x_{1}, …, x_{n}) q_{1}^{(x_{1},x_{1})} … q_{n}^{(x_{n},x_{n})} (q_{i} = e^{2π\sqrt{−1} z_{i}})

and for P ∈ P_{a}(d) this is a modular form of weight d/2 + a_{i} in z_{i} by the classical Hecke-Schoeneberg theory of theta series with harmonic polynomial coefficients. A deeper direct application of our theory to critical values of triple L function is found in [10] (see also [5]) and another direct application is given in [3]. There are various other theories of differential operators and applications of a similar kind (for example, see [8], [9], [11]).

(iii) The spaces P_{a}(d) can be looked at from the point of view of representation theory. Specifically, the map P ↦ P̃ gives an isomorphism between P_{a}(d) and the space of “invariant harmonic polynomials,” i.e., polynomials in n variables x_{i} ∈ ℝ^{d} which are harmonic and homogeneous (of degree a_{i}) in each x_{i} and invariant under the simultaneous action of O(d), the orthogonal group of the scalar product ( , ), on all x_{i}; this space can be analyzed in terms of the classical representation theory of O(d). However, apart from a brief mention of this in connection with the dimension formula proved in §2, we will not use this interpretation, but will work directly with the functions P on S_{n}, since this is more elementary, gives more explicit results, and is applicable in greater generality.

(iv) If n = 3, then the dimension of P_{a}(d) is always 0 or 1, so that the higher spherical polynomials are particularly canonical in this case. The higher spherical polynomials of varying multidegree give an orthogonal basis of L^{2} functions in three variables with respect to a certain scalar product, and the coefficients of these polynomials also turn out to be combinatorially very interesting expressions, including as special cases the classical 3j-symbol (or Clebsch-Gordan coefficients) of quantum mechanics.

(v) In a sequel to this paper [12], we will show that the system of differential equations defining the higher spherical polynomials (generalized Legendre differential equations) in the case n = 3 is equivalent to an integrable Pfaffian system of rank 8, so that there are precisely eight linearly independent solutions, generalizing the Legendre functions of the first and second kinds in the classical case n = 2. In the general theory of differential equations, such systems are very rare: systems of linear partial differential equations with polynomial coefficients generally have a solution space which is either zero- or infinite-dimensional, and it is non-trivial to construct Pfaffian systems satisfying the necessary integrability conditions. The system arising from the study of higher spherical polynomials is therefore also interesting from this point of view.

The contents of the paper are as follows. Chapter I describes the general theory, for arbitrary n and d. In the first two sections, which are basic for everything that follows, we write down the differential equations defining the space P_{n}(d) (now for any d ∈ ℂ), prove a formula for the dimensions of its homogeneous pieces P_{a}(d), and construct a specific basis P_{ν ν ν}^{M} (“monomial basis”) characterized by the property that its restriction to the subspace t_{11} = ⋯ = t_{nn} = 0 consists of monomials. We also give a brief description of the behavior of P_{n}(d) at certain special values of d (in 2ℤ_{≤0}) where its dimension changes. Sections 3 and 4 contain the definition of a scalar product with respect to which the various spaces P_{a}(d) are mutually orthogonal, generalizing one of the basic properties of the classical polynomials for n = 2, and of a Lie algebra 𝔤, isomorphic to 𝔰𝔭(n, ℝ), of differential operators on polynomials on S_{n}. This is then applied in §5 to construct two further canonical bases of P_{n}(d), the “ascending” and the “descending” basis, whose elements are inductively defined by the property that two adjacent basis elements are related by the action of a specific element of the universal enveloping algebra of 𝔤. The first of these two bases is proportional in each fixed multidegree to the monomial basis, and the two bases are mutually dual with respect to the canonical scalar product. Finally, in §6 we study the situation when d is a positive integer less than n and describe the relationship between the spaces P_{n}(d) and {P | P̃ is harmonic in each x_{i}}, which now no longer coincide, in this case.

Chapter II is devoted to explicit constructions of higher spherical polynomials. In §7 we use the relationship between O(4) and GL(2) to construct specific elements of P_{n}(d) when d = 4 and show that they can be obtained as the coefficients of an explicit algebraic generating function. The next section, which is quite long, concerns the case n = 3. This case is especially interesting both because it is the first one beyond the classical case and because the dimension of P_{a}(d) here is always 0 or 1, so that the polynomials are especially canonical. Specifically, the dimension is 1 if and only if a_{1} + a_{2} + a_{3} − 2 max{a_{1}, a_{2}, a_{3}} is even and non-negative, i.e., if we can write

a_{1} = ν_{2} + ν_{3},  a_{2} = ν_{1} + ν_{3},  a_{3} = ν_{1} + ν_{2}  (3)

for some triple ννν = (ν_{1}, ν_{2}, ν_{3}) of integers ν_{i} ≥ 0, which we then call the index of the polynomial. In Subsection A we generalize the generating function given for d = 4 in §7 (now specialized to n = 3) to construct an explicit generating function for n = 3 and generic values of d ∈ ℂ whose coefficients give polynomials P_{ν ν ν} generating the 1-dimensional space P_{a}(d). The next three subsections give explicit formulas for the scalar products of these special polynomials, their relations to the three general bases (monomial, ascending and descending), the recursion relations satisfied by their coefficients, and for the coefficients themselves, while Subsection E discusses the geometry and symmetry of the 3-dimensional space on which these polynomials naturally live. Section 9 contains the construction of generating functions for arbitrary n and d whose coefficients span the spaces of higher spherical polynomials of any given multidegree (and in fact turn out to be proportional to the elements of the descending basis). We also describe how these coefficients can be obtained inductively by the repeated application of certain differential operators and give a number of special cases and explicit formulas. Finally, in §10 we study the bases of higher spherical polynomials coming from these generating functions in the case n = 4 and show that the generating functions themselves, which were given by algebraic expressions for n ≤ 3, now usually are not algebraic, but that for even integral values of d ≥ 4 they become algebraic after being differentiated a finite number of times. The paper ends with some tables of higher spherical polynomials and their scalar products for 3 ≤ n ≤ 5, revealing many interesting regularities and irregularities of these objects.

A list of the principal notations used (omitting standard ones like ℕ = ℤ_{>0}) is given at the end of the paper. Theorems are numbered consecutively throughout the paper, but propositions, proposition-definitions, lemmas and corollaries are numbered separately in each section, and not numbered at all if there is only one of them in a given section. Readers who are primarily interested in the n = 3 case can read just §8, in which we have repeated a few definitions and notations so that it can be read independently of the preceding sections.

Finally, we should perhaps also say a few words about the history of this paper, since it has been in preparation for a very long time and earlier versions circulated for several years and were quoted by several authors. The project originally started well over twenty years ago with a letter sent by the second author to B. Gross and S. Kudla in connection with a plan (later abandoned) to compute the arithmetic height pairings of modularly embedded curves in the modular 3-fold (ℌ/SL(2, ℤ))^{3}. This letter concerned spherical polynomials on GL(2)\GL(2)^{3}/GL(2), corresponding to the case n = 3, d = 4 of the present paper, and gave for that case the generating function for the polynomials and some explicit calculations of their coefficients (now given in Sections 7 and 8). During the second author’s visits to Kyushu in the early 1990’s the first author suggested the formulation in terms of orthogonal groups and the generalization to arbitrary n and d, developed a more general theory (later published separately in [7]), and obtained a number of the results given in this paper, including the generating function for n = 3 and arbitrary d. The preliminary version mentioned above was primarily concerned with the case n = 3 and corresponded to parts of Sections 1–3 and all of Sections 7 and 8, while the remaining results in Sections 2 and 3 and all the material in Sections 4, 5, 6, 9 and 10 was found and written up much later. We delayed publication for many years as new results for higher n emerged, in order to be able to present a more complete and coherent picture.

The first author would like to thank the Max-Planck-Institut für Mathematik in Bonn and the Collège de France in Paris, and the second author the mathematics departments of Kyushu University and Osaka University, for their repeated hospitality during the period when this work was being carried out.

Chapter I. General Theory

§1. The generalized Legendre differential equation

The definition of P_{a}(d) which was given in the introduction makes sense only if d is a positive integer and gives a finite-dimensional space only if d ≥ n, since for d < n the map P ↦ P̃ is not injective (because the (x_{i}, x_{j}) are then algebraically dependent, the relations being the vanishing of all (d + 1) × (d + 1) minors of T). In this section we generalize the definition of P_{a}(d), first to d < n and then even to complex values of d, by thinking of the elements of P_{a}(d) as polynomial solutions of a certain system of differential equations generalizing the classical Legendre equation. This system of differential equations will be used in §2 to calculate the dimension of P_{a}(d). As illustrations of the general situation we discuss the two examples n = 2, a = (a, a), and n = 4, a = (2, 2, 2, 2) in some detail. At the end of the section we also give a brief discussion of the inhomogeneous version of the higher spherical polynomials and their differential equations.

First, we must say something about the coordinates on the space S_{n}. Of course we could use the n(n + 1)/2 independent numbers t_{ij} with 1 ≤ i ≤ j ≤ n, but this does not respect the symmetry. Instead, we will use all n^{2} components t_{ij} of T ∈ S_{n} as variables, with t_{ij} = t_{ji}, and will write elements^{1} of ℂ[S_{n}] symmetrically as polynomials in the variables \sqrt{t_{ij}t_{ji}} (= t_{ij}), i.e. we take as a basis for ℂ[S_{n}] the set of monomials T^{ν ν ν} := ∏_{i,j=1}^{n} t_{ij}^{ν_{ij}/2} with ννν ranging over the set

N = N_{n} = {ννν = (ν_{ij})_{1≤i,j≤n} | ν_{ij} = ν_{ji} ∈ ℤ_{≥0} , ν_{ii} ≡ 0 (mod 2)}

of even symmetric n × n matrices with non-negative entries. There is a canonical isomorphism ℂ[S_{n}] ≅ ℂ^{N} given by mapping a polynomial ∑ C_{ν ν ν} T^{ν ν ν} to its set of coefficients {C_{ν ν ν}}. For the differentiation operator with respect to t_{ij} we take ∂_{ij} = (1 + δ_{ij})∂/∂t_{ij}. (Note: We had to do something similar in eq. (2) for the same reason, namely, that the coordinates z_{ij} are not independent variables. There we also included a normalizing factor 1/2, because for Siegel modular forms the diagonal elements are the most important ones and one wants a natural generalization of n = 1, but in our situation the non-diagonal entries of the matrix are equally important and we have chosen a normalization which preserves the lattice of polynomials with integral coefficients.) The action of ∂_{ij} on our chosen basis of ℂ[S_{n}] is given by T^{ν ν ν} ↦ ν_{ij} T^{ν ν ν−e_{ij}}. Here e_{ij} denotes the n × n matrix with (a, b)-component δ_{ia}δ_{jb} + δ_{ib}δ_{ja}(i.e., the matrix with (i, j) and (j, i) entries equal to 1 and all other entries equal to 0 if i ≠ j, and with (i, i)-entry 2 and all other entries 0 if i = j), so that N is the free abelian semigroup generated by the e_{ij}.

We define the multidegree of the monomial T^{ν ν ν} to be the vector ννν·1 ∈ ℤ_{≥0}^{n}, where 1 is the vector of length n with all components equal to 1, and denote by ℂ[S_{n}]_{a} the subspace of ℂ[S_{n}] spanned by all monomials of given multidegree a. Thus

ℂ[S_{n}]_{a} ≅ ℂ^{N (a)} ( N (a) := {ννν ∈ N | ννν·1 = a} )

with respect to the above isomorphism ℂ[S_{n}] ≅ ℂ^{N}, while more intrinsically

ℂ[S_{n}]_{a} = {P ∈ ℂ[S_{n}] | P (λT λ) = λ^{a}P (T) for all λ ∈ diag(ℂ^{n})} ,  (4)

where λ^{a} = ∏ λ_{i}^{a_{i}} for λ = diag(λ_{1}, …, λ_{n}). For P ∈ ℂ[S_{n}]_{a} and (ℝ^{d}, ( , )) as in the introduction, the function P̃ : (ℝ^{d})^{n} → ℂ has multidegree a . Hence, if d ≥ n (so that the map P ↦ P̃ is injective), then the space P_{a}(d) as defined in the introduction is a subspace of the space ℂ[S_{n}]_{a}. Specifically, it is the space of all polynomials P ∈ ℂ[S_{n}]_{a} for which the polynomial P̃ (which is now automatically multihomogeneous of multidegree a in the x’s) is harmonic with respect to each x_{i}. We would like to define P_{a}(d) for all values of d as a subspace of P ∈ ℂ[S_{n}]_{a}. To do this, we must express the condition that P̃ is multiharmonic directly in terms of the polynomial P. The answer will be a system of differential equations with coefficients depending polynomially on d; these will then make sense for any complex number d.

1Abusing notation slightly, we will use K[ℝ^{d}] or K[ℂ^{d}] (or more generally K[V]) to denote the algebra of polynomials in d variables (or more generally of polynomials on an arbitrary vector space V) with coefficients in a field K.

The calculation is straightforward. The harmonicity conditions are Δ_{i}P̃ = 0 for 1 ≤ i ≤ n, where Δ_{i} denotes the Laplacian^{2} with respect to the variable x_{i} ∈ ℝ^{d} (and with respect to the chosen scalar product in ℝ^{d}). Let x_{iα} (α = 1, …, d) be the coordinates of x_{i} in a coordinate system for which the scalar product (and hence also the Laplacian) is the standard one. Then t_{ij} = ∑_{α} x_{iα}x_{jα}, so

\frac{∂P̃}{∂x_{iα}} = ∑_{j=1}^{n} x_{jα} (∂_{ij}P )^{∼} ,

with ∂_{ij} defined as above. Applying this formula a second time, we get

\frac{∂^{2}P̃}{∂x_{iα}^{2}} = (∂_{ii}P )^{∼} + ∑_{j,k=1}^{n} x_{jα} x_{kα} (∂_{ij}∂_{ik}P )^{∼} ,

and summing this over α gives Δ_{i}(P̃) = (D_{i}P)^{∼}, where

D_{i} = D_{i}^{(d)} = d ∂_{ii} + ∑_{j,k=1}^{n} t_{jk} ∂_{ij}∂_{ik}  (1 ≤ i ≤ n) .  (5)

(Recall that ∂_{ii} = 2 ∂/∂t_{ii}.) We therefore have the following:

Proposition-Definition. For n ∈ ℕ, a ∈ ℤ_{≥0}^{n}, and arbitrary d ∈ ℂ, set

P_{a}(d) = {P ∈ ℂ[S_{n}]_{a} | D_{1}P = … = D_{n}P = 0}

with D_{i} = D_{i}^{(d)} as in (5). For integral d ≥ n, this agrees with the definitions given in the introduction.

Remark. The space ℂ[S_{n}]_{a} is itself the solution set of the system of linear differential equations E_{i}P = a_{i}P (1 ≤ i ≤ n), where E_{i} is the Euler operator ∑_{j=1}^{n} t_{ij}∂_{ij}. Thus one

could also define P_{a}(d) as the subspace of ℂ[S_{n}] annihilated by the operators D_{i}^{(d)} and E_{i} − a_{i} for 1 ≤ i ≤ n. We also observe that the spaces ℂ[S_{n}]_{a} and P_{a}(d) are 0 unless the integer a·1 = a_{1} + ⋯ + a_{n} (total degree) is even. From now on we will usually assume this.

Since the space P_{a}(d) is always a subset of ℂ[S_{n}]_{a}, its dimension for any d ∈ ℂ is at most N(a) := #N (a), the number of nonnegative even symmetric matrices with row sums a_{1}, …, a_{n}. We will show in §2 that (except possibly for d belonging to a finite set of nonpositive even integers) the dimension of P_{a}(d) is independent of d and equals N_{0}(a), the cardinality of the set N_{0}(a) = {ννν ∈ N (a) | ν_{ii} = 0 (∀i)}. We illustrate this with two examples.

Example 1: n = 2 . In this case the operator t_{11}D_{1}^{(d)} − t_{22}D_{2}^{(d)} acts on polynomials of bidegree (a_{1}, a_{2}) as multiplication by (a_{1} − a_{2})(a_{1} + a_{2} + d − 2), so if d is a positive integer, or indeed any complex number except 2 − a_{1} − a_{2}, then P_{a}(d) can be non-zero only if a = (a, a) for some a ∈ ℤ_{≥0}, as was already mentioned in §1. If a = (a, a), then N (a) is the set of matrices

2Note that, since everything is purely algebraic, we could also work over ℂ rather than ℝ, but the condition for an element of ℂ[(ℂ^{d})^{n}] to be harmonic in the ith variable z_{i} = (z_{i1}, …, z_{id}) ∈ ℂ^{d} would still be with respect to the complexified real Laplacian ∑_{α} ∂^{2}/∂z_{iα}^{2}, not the complex Laplacian ∑_{α} ∂^{2}/∂z_{i,α}∂z̄_{i,α}.

( \begin{matrix}2l & a − 2l \\ a − 2l & 2l \end{matrix} ) with 0 ≤ l ≤ a/2, so N(a) = [a/2] + 1, while N_{0}(a) contains only the matrix

( \begin{matrix}0 & a \\ a & 0 \end{matrix} ), so N_{0}(a) = 1. If P (T) = ∑_{l=0}^{[a/2]} c_{l}(t_{11}t_{22})^{l}(t_{12}t_{21})^{(a−2l)/2} is a polynomial in ℂ[S_{2}], then

we find after a short calculation that each of the two equations D_{1}^{(d)}P = 0 and D_{2}^{(d)}P = 0 is equivalent to the recursion

4l(a − l + \frac{d}{2} − 1) c_{l} + (a − 2l + 2)(a − 2l + 1) c_{l−1} = 0 (1 ≤ l ≤ [a/2] )

and that this recursion has a one-dimensional space of solutions for every complex number d, even though the recursion itself behaves slightly differently for certain exceptional values of d (namely, the even integers in the interval [4 − 2a, 2 − a]). Thus the dimension of P_{a}(d) is always equal to N_{0}(a) = 1, as claimed, with the generator of the space being the homogeneous form of a Gegenbauer polynomial (respectively a Legendre polynomial if d = 3 or a Chebyshev polynomial of the first or second kind if d = 2 or 4). We also see in this example that the dimension formula dim P_{a}(d) = N_{0}(a) fails in the exceptional case d = 2 − a_{1} − a_{2}, the simplest example being a = (2, 0) and d = 0, when P_{a}(d) contains the polynomial t_{11} and hence has dimension N(a) = 1, whereas N_{0}(a) = 0. In fact, P_{(a_{1},a_{2})}(2−a_{1} −a_{2}) has dimension 1 for any a_{1}, a_{2} ≥ 0 with a_{1} ≡ a_{2} (mod 2), while N_{0}(a_{1}, a_{2}) = 0 unless a_{1} = a_{2}.

Example 2: n = 4, a = (2, 2, 2, 2) . In this case N(a) = 17, a basis for ℂ[S_{n}]_{a} being given by the monomials t_{11}t_{22}t_{33}t_{44}, t_{11}t_{22}t_{34}^{2}, t_{11}t_{23}t_{34}t_{42}, t_{13}t_{14}t_{23}t_{24}, t_{12}^{2}t_{34}^{2} and their permutations (1 + 6 + 4 + 3 + 3 = 17). The system of differential equations D_{1}P = D_{2}P = D_{3}P = D_{4}P = 0 imposes 4 × 5 = 20 conditions. (For instance, D_{1} maps P to the 5-dimensional space spanned by t_{22}t_{33}t_{44}, t_{22}t_{34}^{2}, t_{33}t_{24}^{2}, t_{44}t_{23}^{2}, and t_{23}t_{34}t_{42}.) A priori one would not expect a system of 20 linear equations in 17 unknowns to have any solutions at all, but our claim dim P_{a}(d) = N_{0}(a) implies that there should in fact be N_{0}(a) = 6 linearly independent ones, i.e. the rank of the 20 × 17 matrix of coefficients should be only 11. Writing down this matrix, we find that this is indeed the case and that the solution space is spanned by the following:

i) (t_{11}t_{22} − d t_{12}^{2})(t_{33}t_{44} − d t_{34}^{2}), (t_{11}t_{33} − d t_{13}^{2})(t_{22}t_{44} − d t_{24}^{2}), (t_{11}t_{44} − d t_{14}^{2})(t_{22}t_{33} − d t_{23}^{2}) , ii) t_{12}t_{34}(t_{13}t_{24} − t_{14}t_{23}), t_{14}t_{23}(t_{12}t_{34} − t_{13}t_{24}), t_{13}t_{24}(t_{14}t_{23} − t_{12}t_{34}) ,

iii) t_{11}t_{23}t_{34}t_{42} + t_{22}t_{13}t_{34}t_{41} + t_{33}t_{12}t_{24}t_{41} + t_{44}t_{12}t_{23}t_{31} − (t_{12}^{2}t_{34}^{2} + t_{13}^{2}t_{24}^{2} + t_{14}^{2}t_{23}^{2}) − d t_{12}t_{13}t_{24}t_{34 .}

(This list actually contains seven solutions, but the three in group (ii) have sum zero and we have given all three only to preserve the symmetry, while the solution (iii), although written asymmetrically, is symmetric in i = 1, …, 4 modulo linear combinations of the solutions (ii).)

We end this section by saying something about the inhomogeneous versions of our higher spherical polynomials. In the classical n = 2, the polynomials in P_{a}(d) (here a = (a, a) for some integer a ≥ 0) are interpreted as polynomials of one variable via the correspondence

P ∈ P_{(a,a)}(d) ⇒ P ( \begin{matrix}t_{11} & t_{12} \\ t_{12} & t_{22} \end{matrix} ) = (t_{11}t_{22})^{a/2} p(\frac{t_{12}}{\sqrt{t_{11}t_{22}}} ) , p(t) = P ( \begin{matrix}1 & t \\ t & 1 \end{matrix} ) ,

in which case the two differential equations defining P_{a}(d) are equivalent to a single differential equation (Legendre or Gegenbauer differential equation) for the polynomial p(t). In the same way, for n > 2 we can pass back and forth via homogenization and dehomogenization between the spaces of multihomogeneous polynomials of multidegree a on S_{n} and polynomials of multidegree ≤ a and multi-parity a on S_{n}^{1} = {T ∈ S_{n} | t_{ii} = 1 (∀ i)}, the correspondence being given by

P ∈ ℂ[S_{n}]_{a} ↔ p = P |_{S_{n}^{1}} ∈ ℂ[S_{n}^{1}] , P (T) = δ(T)^{a/2} p(T^{(1)}) ,

where δ(T)^{a/2} = ∏_{i=1}^{n} t_{ii}^{a_{i}/2} , T^{(1)} = (τ_{ij}) , τ_{ij} = \frac{t_{ij}}{\sqrt{t_{ii}t_{jj}}} .

(6)

(The conditions on the degree and parity of p ensure that the function of T defined by the expression on the right is a polynomial.) Just as in the classical case, we have (eq. (10) below) that every polynomial on S_{n}^{1} is the restriction of a unique higher spherical polynomial (element of P_{n}(d)). More interestingly, the system of differential equations in n(n+1)/2 variables defining the space P_{a}(d) will correspond to a system of n diffential equations in the n(n−1)/2 coordinates of S_{n}^{1}. For instance, for n = 3, if we define coordinates t_{i} on S_{3}^{1} by

S_{3}^{1} ∋ T = ( \begin{matrix}1 & t_{3} & t_{2} \\ t_{3} & 1 & t_{1} \\ t_{2} & t_{1} & 1 \end{matrix} )  (7)

(so that t_{1} = τ_{23} etc.), then the condition on a polynomial Q(t_{1}, t_{2}, t_{3}) to be the restriction to S_{3}^{1} of an element of P_{a}(d) is that Q satisfy the system of differential equations given by

\begin{matrix}(1 − t_{2}^{2}) \frac{∂^{2}Q}{∂t_{2}^{2}} + 2 (t_{1} − t_{2}t_{3}) \frac{∂^{2}Q}{∂t_{2}∂t_{3}} + (1 − t_{3}^{2}) \frac{∂^{2}Q}{∂t_{3}^{2}} \\ − (d − 1) (t_{2} \frac{∂Q}{∂t_{2}} + t_{3} \frac{∂Q}{∂t_{3}}) + a_{1}(a_{1} + d − 2) Q = 0 \end{matrix}  (8)

and its two cyclic permutations. This system of differential equations will be studied in detail in [12], while some special properties of the coordinates (t_{1}, t_{2}, t_{3}) defined by (7) for the case n = 3 will be discussed in §8.E of the present paper.

§2. Dimension formula and decomposition theorem

Let N (a) and N_{0}(a) ⊂ N (a) be as in the last section and Φ : P_{a}(d) → ℂ^{N_{0}(a)} the natural projection map (the composite of the inclusion P_{a}(d) ⊆ ℂ[S_{n}]_{a} ≅ ℂ^{N (a)} with the projection map ℂ^{N (a)} → ℂ^{N_{0}(a)}), which corresponds to the operation of restricting a polynomial P (T) to the subspace S_{n}^{0} ⊂ S_{n} of matrices T whose diagonal coefficients vanish. The main result of this section is:

Theorem 1. Let a ∈ ℤ_{≥0}^{n} be a multidegree. Then for any complex number d not belonging to the finite subset

Ξ(a) := 2ℤ ∩ ⋃_{a_{i}≥2}[4 − 2a_{i}, 2 − a_{i}]

of 2 ℤ_{≤0}, the map Φ : P_{a}(d) → ℂ^{N_{0}(a)} is an isomorphism. In particular, the dimension of P_{a}(d) for such d equals N_{0}(a), the cardinality of N_{0}(a).

Proof. Let P (T) = ∑_{ν ν ν} C(ννν)T^{ν ν ν} be an element of P_{a}(d), i.e. P is annihilated by each of the operators D_{i} = D_{i}^{(d)} defined in (5). The action of D_{i} on monomials is given by

D_{i}(T^{ν ν ν}) = ν_{ii} (2a_{i} − ν_{ii} + d − 2) T^{ν ν ν−e_{ii}} + ∑_{j,k≠i} ν_{ij}(ν_{ik} − δ_{jk}) T^{ν ν ν−e_{ij}−e_{ik}+e_{jk}} ,

where the sum runs over j, k ∈ {1, …, î, …, n} and e_{ab} has the same meaning as in §2, and using this we can rewrite the differential equation D_{i}P = 0 as a recursion formula for the coefficients, as we did for the example n = 2 in the last section. This recursion for i = 1, together with the assumption that 2a_{1} − ν_{11} + d − 2 does not vanish for any even integer ν_{11} ∈ (0, a_{1}], shows that the coefficients C(ννν) with ν_{11} > 0 can be expressed as linear combinations of those with a smaller value of ν_{11} and hence that P is determined by the C(ννν) with ν_{11} = 0. Similarly, the equation D_{2}P = 0 shows that the coefficients C(ννν) with ν_{11} = 0 and ν_{22} > 0 are combinations of C(ννν′) with ν′_{11} = 0 and ν′_{22} < ν_{22}, so that they are determined by the coefficients with ν_{22} = 0. Continuing in this way, we see by induction that the coefficients C(ννν) with ννν ∈ N_{0}(a) determine P completely. Hence Φ on P_{a}(d) is injective and dim P_{a}(d) ≤ N_{0}(a).

To show directly that Φ is also surjective, we would have to show that the system of equations D_{i}P = 0 can be solved for any choice of the coefficients C(ννν) with ννν ∈ N_{0}(a). But this is not at all obvious, as the examples in §2 make clear. Instead, we will prove by induction on i the stronger statement that the dimension of K_{i} = K_{i}(a) := Ker(D_{1}) ∩ ⋯ ∩ Ker(D_{i}) is exactly N^{(i)}(a) for each i between 0 and n, where N^{(i)}(a) is the number of matrices in N (a) with ν_{11} = … = ν_{ii} = 0. The case i = n gives the assertion we want. The main observation for the proof is that the various operators D_{i} commute. This is obvious when d is a positive integer because of the interpretation of the D_{i}’s as the Laplacians with respect to disjoint sets of variables x_{i} ∈ ℝ^{d}, and it then remains true for all d because the coefficients of the operators are polynomials in d and no polynomial has infinitely many zeros. It follows that D_{i} maps K_{i−1} to K_{i−1} or more precisely, since D_{i} clearly decreases a = ννν · 1 by 2e_{i} (where e_{i} as before denotes the vector with 1 in the ith place and 0’s elsewhere), K_{i−1}(a) to K_{i−1}(a − 2e_{i}). If we therefore assume inductively that dim K_{i−1}(a) = N^{(i−1)}(a) for all a, then it follows that

dim K_{i}(a) = dim Ker( D_{i} : K_{i−1}(a) → K_{i−1}(a − 2e_{i}))
≥ dim K_{i−1}(a) − dim K_{i−1}(a − 2e_{i})
= N^{(i−1)}(a) − N^{(i−1)}(a − 2e_{i}) = N^{(i)}(a) ,

and since the previous argument gives the reverse inequality, this completes the proof. □

Remark 1. The definitions of N(a) and N_{0}(a) are equivalent to the generating functions

∑_{a≥0} N(a) z_{1}^{a_{1}} ⋯ z_{n}^{a_{n}} = ∏_{1≤i≤j≤n} \frac{1}{1 − z_{i}z_{j}} ,  ∑_{a≥0} N_{0}(a) z_{1}^{a_{1}} ⋯ z_{n}^{a_{n}} = ∏_{1≤i<j≤n} \frac{1}{1 − z_{i}z_{j}} .

Remark 2. There are simple expressions for N_{0}(a) if n ≤ 4. First note that, for any n, a necessary condition for N_{0}(a) ≠ 0 is that the sum of the a_{i}’s is even and that no a_{i} is strictly bigger than the sum of the others, i.e., that the number δ := \frac{1}{2} ∑_{i} a_{i} − max_{i} a_{i} is a nonnegative

integer. Assuming that a satisfies this condition, we have the following formulas for n ≤ 4:

N_{0}(a_{1}, a_{2}) = 1,  N_{0}(a_{1}, a_{2}, a_{3}) = 1,  N_{0}(a_{1}, a_{2}, a_{3}, a_{4}) = \frac{(s + 1)(s + 2)}{2} ,

where s in the last case is min{a_{1}, a_{2}, a_{3}, a_{4}, δ}. Note that for n = 2 and n = 3 the condition δ ≥ 0 is equivalent to a_{1} = a_{2} and to the triangle inequality, respectively. For general values of n, there is no simple formula for N_{0}(a), but assuming that n ≥ 4 and that no a_{i} vanishes, we have N_{0}(a) = 1 if and only if δ = 0. Under the same assumption, one has the following partial results:

\begin{matrix}δ & N_{0}(a) \\ < 0 & 0 \\ 0 & 1 \\ 1 & \binom{n−1}{2} \\ 2 & \binom{M+1}{2} where M = \binom{n−1}{2} − #\{i | a_{i} = 1\} \\ ≥ 3 & n^{2δ}/2^{δ}δ! + O(n^{2δ−1}) \end{matrix}

Notice that the dimension is a triangular number whenever n ≤ 4 or δ ≤ 2, but this is not true in general, the smallest counterexample being N_{0}(2, 2, 2, 2, 2) = 22.

Remark 3. As mentioned in the introduction, in the case where d is an integer ≥ n, one can also see the dimension formula via the interpretation of P_{a}(d) as the O(d)-invariant subspace of H_{a_{1}}(ℝ^{d}) ⊗ … ⊗ H_{a_{n}}(ℝ^{d}), where H_{a}(ℝ^{d}) denotes the space of homogeneous harmonic polynomials of degree a in ℝ^{d} and O(d) acts diagonally on (ℝ^{d})^{n}. Let ρ_{a_{i}} be the symmetric tensor representation of GL(d) of degree a_{i}, that is, the representation on degree a_{i} homogeneous polynomials. We denote by χ_{a_{i}} the character of ρ_{a_{i}}. Now consider the tensor product representation ⊗_{i=1}^{n}ρ_{a_{i}} of GL(d). We take the restriction of this representation to O(d). For d ≥ n, the isotypic component of the trivial representation is spanned by the linearly independent polynomials ∏_{i,j=1}^{n}(x_{i}, x_{j})^{ν_{ij}/2} with ν_{ij} ∈ ℤ_{≥0}, ν_{ii} ∈ 2ℤ and a_{i} = ∑_{j=1}^{n} ν_{ij}, by classical invariant theory (Weyl [16], pages 53 and 75). Their number is exactly N(a). On the other hand, the character of the representation of O(d) on harmonic polynomials of degree a_{i} is χ_{a_{i}} − χ_{a_{i}−2}. What we want to know is the multiplicity of the trivial representation of O(d) in ∏_{i=1}^{n}(χ_{a_{i}} − χ_{a_{i}−2}). Hence, expanding the parenthesis and counting the multiplicities of the trivial character for each product, we can see that the multiplicity in ∏_{i=1}^{n}(χ_{a_{i}} − χ_{a_{i}−2}) is the coefficient of ∏ z_{i}^{a_{i}} in

∏_{i=1}^{n}(1 − z_{i}^{2}) ∏_{1≤i≤j≤n}(1 − z_{i}z_{j})^{−1} = ∏_{1≤i<j≤n}(1 − z_{i}z_{j})^{−1} ,

as asserted.

Remark 4. From the proof of Theorem 1 we deduce the analogue for the operators D_{i} of the classical fact that any collection of functions f_{i}(x_{1}, …, x_{n}) (i = 1, …, n) satisfying ∂f_{i}/∂x_{j} = ∂f_{j}/∂x_{i} for all i and j is the collection of partial derivatives ∂g/∂x_{i} of a single function g. Here the assertion is that, if d ∉ Ξ(a), then any collection of functions P_{i} ∈ ℂ[S_{n}]_{a−2e_{i}} (i = 1, …, n) satisfying D_{i}(P_{j}) = D_{j}(P_{i}) for all i and j has the form P_{i} = D_{i}(Q) for some Q ∈ ℂ[S_{n}]_{a}. Indeed, the proof of Theorem 1 implies (in the notations there) that the map D_{i} : K_{i−1}(a) → K_{i−1}(a − 2e_{i}) is surjective for all i. Applying this with i = 1 (where K_{0}(a) = ℂ[S_{n}]_{a} and K_{0}(a − 2e_{1}) = ℂ[S_{n}]_{a−2e_{1}}), we can find Q_{1} ∈ ℂ[S]_{a} such that D_{1}Q_{1} = P_{1}. Then D_{1}(P_{2} − D_{2}Q_{1}) = D_{2}P_{1} − D_{2}P_{1} = 0, so P_{2} − D_{2}Q_{1} ∈ K_{1}(a − 2e_{2}). Now the surjectivity of D_{2} : K_{1}(a) → K_{1}(a − 2e_{2}) implies that there is a Q_{2} ∈ ℂ[S_{n}]_{a} with D_{1}Q_{2} = 0 and D_{2}Q_{2} = P_{2} − D_{2}Q_{1}, or equivalently D_{1}(Q_{1} + Q_{2}) = P_{1} and D_{2}(Q_{1} + Q_{2}) = P_{2}. Continuing in this way, we obtain the statement claimed, which will be used again in the Remark following Proposition 2 below and at several later points in the paper.

Theorem 1 has two interesting consequences. The first one, which is a refinement of the dimension statement, is an analogue of the classical fact that every polynomial on ℝ^{d} has a unique decomposition into summands of the form (x, x)^{j}P_{j}(x) with each P_{j}(x) harmonic.

Corollary. For any d ∈ ℂ ∖ 2ℤ_{≤0} we have

ℂ[S_{n}] = P^{(n)}(d) [t_{11}, t_{22}, …, t_{nn}] .

More precisely, for a and d be as in Theorem 1, the space ℂ[S_{n}]_{a} of all polynomials on S_{n} of multidegree a has a direct sum decomposition as

ℂ[S_{n}]_{a} = ⨁_{0≤m≤a/2} δ(T)^{m} P_{a−2m}(d) ,  (9)

where the sum ranges over all m ∈ ℤ^{n} satisfying 0 ≤ m_{i} ≤ a_{i}/2 and δ(T)^{m} = ∏_{i=1}^{n} t_{ii}^{m_{i}} .

Proof. The proof of Theorem 1 shows that the map Φ_{i} : K_{i}(a) → ℂ^{N^{(i)}(a)}, where N^{(i)}(a) is the set of ννν ∈ N (a) with ν_{11} = ⋯ = ν_{ii} = 0, is injective (i.e., any P ∈ K_{i}(a) is determined by its coefficients C(ννν) with ννν ∈ N^{(i)}(a)) and surjective (for dimension reasons), and that the sequence 0 → K_{i}(a) → K_{i−1}(a) →^{D_{i}} K_{i−1}(a − 2e_{i}) → 0 is exact. Now let P (T) be an arbitrary element of ℂ[S_{n}]_{a} = ℂ^{N (a)}. Its canonical projection to ℂ^{N^{(1)}(a)} equals Φ_{1}(Q_{0}) for a unique element Q_{0} of K_{1}(a). But this means that the restrictions of P and Q_{0} to t_{11} = 0 agree, so we have P (T) = Q_{0}(T) + t_{11}P_{1}(T) for some P_{1} ∈ ℂ[S_{n}]_{a−2e_{1}}, and this decomposition is unique. Applying the same argument to P_{1}, we get P_{1} = Q_{1} + t_{11}P_{2} with Q_{1} ∈ K_{1}(a − 2e_{1}) and P_{2} ∈ ℂ[S_{n}]_{a−4e_{1}}, and continuing in this way we find that P has a unique decomposition of the form P (T) = ∑_{0≤m_{1}≤a_{1}/2} t_{11}^{m_{1}}Q_{m_{1}}(T) with Q_{m_{1}} ∈ K_{1}(a − 2m_{1}e_{1}) for each m_{1}. Now we use the same argument to show that each Q_{m_{1}} can be decomposed uniquely as ∑_{0≤m_{2}≤a_{2}/2} t_{22}^{m_{2}}Q_{m_{1},m_{2}}(T) with Q_{m_{1},m_{2}} ∈ K_{2}(a − 2m_{1}e_{1} − 2m_{2}e_{2}), etc., obtaining in the end precisely the direct sum decomposition asserted by the corollary. □

The decomposition (9), which will be shown in the next section to be orthogonal with respect to a natural scalar product on ℂ[S_{n}], is one of the key properties of higher spherical polynomials. Note that it can also be formulated in terms of the inhomogeneous coordinates discussed in the final paragraph of §1 as the assertion that the restriction of the projection map ℂ[S_{n}] → ℂ[S_{n}^{1}] to the space of higher spherical polynomials is an isomorphism:

ℂ[S_{n}^{1}] = P_{n}(d)|_{S_{n}^{1}} = ⨁_{a} P_{a}(d)|_{S_{n}^{1}} .  (10)

For n = 2 this is just the classical decomposition of polynomials in one variable into linear combinations of Legendre or Gegenbauer polynomials

The second consequence of Theorem 1 which we want to mention is that it gives us canonical bases of the spaces P_{a}(d) and P^{(n)}(d). Let N_{0} = ⋃ N_{0}(a) be the set of all matrices ννν ∈ N with all ν_{ii} = 0 . Then by pulling back the basis of monomials T^{ν ν ν} under the isomorphism Φ of the theorem, we obtain:

Proposition 1. Let a ∈ ℤ_{≥0}^{n} and d ∈ ℂ ∖ Ξ(a). Then for every ν ∈ N_{0}(a) there is a unique element P_{ν ν ν,d}^{M} ∈ P^{(n)}(d) whose restriction to S_{n}^{0} is the monomial T^{ν ν ν}, and these polynomials form a basis of P_{a}(d).

The polynomials P_{ν ν ν}^{M}(T) (the index d will usually be dropped when no confusion can result) will be called the monomial basis of P_{n}(d). In Section 5 we will study this basis (and its dual basis with respect to a canonical scalar product) in more detail. We will also give explicit constructions of these polynomials there. This also gives another proof of Theorem 1 and its corollary.

Finally, let us make a few remarks concerning the degenerate case d ∈ Ξ(a). In this case the dimension of P_{a}(d) can be larger than N_{0}(a), as we saw in Example 1. However, there is a natural way to define a subspace P_{a}^{∗}(d) of P_{a}(d) of the correct dimension, because it is not hard to see that the subspaces P_{a}(d′) ⊂ ℂ[S_{n}] have a limiting value as d′ → d. For example, the six polynomials in P_{(2,2,2,2)}(d) given in Example 2 become linearly dependent if d = 0 (the three polynomials in i) become equal), but we can make a change of basis over ℚ(d) to give six other polynomials, still with coefficients depending polynomially on d, which are linearly independent for all d. (Replace the second and third solutions in i) by their differences with the first solution divided by d.) This means that if we replace the space P_{a}(d) by the limiting space

P_{a}^{∗}(d) := lim_{\begin{matrix}d′→d \\ d′≠d \end{matrix}} P_{a}(d′)  (11)

whenever d belongs to Ξ(a), then the dimension formula dim P_{a}(d) = N_{0}(a) holds for all d ∈ ℂ and the map P_{a}^{∗} from ℂ to the Grassmannian of N_{0}(a)-dimensional subspaces of ℂ[S_{n}]_{a} is continuous, and in fact analytic, everywhere. For this reason, we could equally reasonably have taken P_{a}^{∗}(d) rather than ∩ KerD_{i} as the definition of P_{a}(d) for values of d belonging to Ξ(a). On the other hand, P_{a}^{∗}(d) is not as nicely behaved as P_{a}(d) for generic d: the “constant term” map from P_{a}^{∗}(d) to ℂ^{N_{0}(a)} is no longer an isomorphism, even though the dimensions are equal, and the decomposition (9) also breaks down, as the following examples show. Example 1: If n = 2 and a = (2, 0), then we have P_{a}(d) = 0 for d ≠ 0 and P_{a}(0) = ℂ t_{11}, but we have P_{a}^{∗}(0) = 0. Example 2: If n = 2 and a = (4, 4), then for d ∉ Ξ(a) = {−2, −4}, the space P_{a}(d) is spanned by (d + 4)(d + 2)t_{12}^{4} − 6(d + 2)t_{11}t_{22}t_{12}^{2} + 3t_{11}^{2}t_{22}^{2}, while P_{a}(−2) and P_{a}(−4) are spanned by t_{11}^{2}t_{22}^{2} and 4t_{11}t_{22}t_{12}^{2} + t_{11}^{2}t_{22}^{2}, respectively. So in this case P_{a}(−2) = P_{a}^{∗}(−2) and P_{a}(−4) = P_{a}^{∗}(−4). In particular, for d = −2 and −4, there exists no monomial basis.

We end this section by giving an alternative, purely algebraic, definition of P_{a}^{∗}(d).

Proposition 2. For a ∈ ℤ_{≥0}^{n} and arbitrary d ∈ ℂ the space P_{a}^{∗}(d) defined by (11) coincides with the space of polynomials P belonging to a sequence (…, P_{2}, P_{1}, P_{0} = P, P_{−1} = 0) of polynomials P_{r} ∈ ℂ[S_{n}]_{a} satisfying the system of differential equations

D_{i}(P_{r}) = ∂_{ii}(P_{r−1})  (r = 0, 1, 2, …, 1 ≤ i ≤ n) .  (12)

Moreover, the polynomials P_{r} can be chosen such that only finitely many of them are non-zero.

Proof. By the discussion preceding the statement of the proposition, any P ∈ P_{a}^{∗}(d) can be deformed to a (convergent or formal) power series P (T; ε) ∈ ℂ[S_{n}][[ε]] belonging to P_{a}^{∗}(d−ε) for any (complex or infinitesimal) value of ε. (Choose a basis {P^{(i)}(T, d′)}_{1≤i≤N_{0}(a)} of P_{a}^{∗}(d′) for all d′ near to d, or even for all d′ ∈ ℂ, where each P^{(i)}(T, d′) is a polynomial in T and d′, write P (T) as ∑_{i} α_{i}P^{(i)}(T, d), choose any convergent or formal power series a_{i}(ε) with a_{i}(0) = α_{i}, and set P (T; ε) = ∑_{i} a_{i}(ε)P^{(i)}(T, d − ε).) Write P (T; ε) as ∑_{r≥0} P_{r}(T) ε^{r}. From the definition (5) we get

D_{i}^{(d−ε)}(P (T; ε)) = (D_{i}^{(d)} − ε∂_{ii})(∑_{r=0}^{∞} P_{r}(T) ε^{r}) = ∑_{r=0}^{∞}(D_{i}^{(d)}(P_{r}) − ∂_{ii}(P_{r−1})) ε^{r}

for each 1 ≤ i ≤ n, so the condition P (T; ε) ∈ P_{a}^{∗}(d−ε) is equivalent to the system of differential equations (12). If we choose the power series a_{i}(ε) in the above argument to be polynomials (e.g., constant polynomials) in ε, then P (T; ε) ∈ ℂ[S_{n}][ε] and the polynomials P_{r}(T) vanish for all but finitely many r. □

Remark. In the generic case d ∉ Ξ(a), we can also deduce the existence of polynomials P_{r}(T) as above for any P ∈ P_{a}(d) by using the fact given in Remark 4 above. Indeed, if we set P_{−1} = 0 and P_{0} = P, then (12) is satisfied for r = 0 by the definition of P_{a}(d). Now suppose by induction that for some R ≥ 0 we have found polynomials (P_{R}, …, P_{0} = P, P_{−1} = 0) satisfying equation (12) for 0 ≤ r ≤ R. Then from the commutation relation

[D_{i}, ∂_{jj}] = ∑_{k,ℓ} [t_{kℓ}, ∂_{jj}] ∂_{ik}∂_{iℓ} = −2 ∂_{ij}^{2}

we find that D_{i}(∂_{jj}(P_{R})) = ∂_{jj}∂_{ii}(P_{R−1}) − 2∂_{ij}^{2}(P_{R}), which is symmetric in i and j, so the result of Remark 4 implies there exists some P_{R+1} ∈ ℂ[S_{n}]_{a} with D_{j}(P_{R+1}) = ∂_{jj}(P_{R}) for all j, completing the induction. We also observe that the freedom in choosing each new polynomial P_{r} (r ≥ 1) in the inductive system (12) is precisely P_{a}(d), corresponding to the freedom of the choice of a_{i}(ε) ∈ α_{i} + ε ℂ[[ε]] in the proof above.

§3. Scalar product and orthogonality

One of the most important properties of the classical Legendre and Gegenbauer polynomials is that they are orthogonal with respect to a suitable scalar product. In this section we prove an analogous property for the higher spherical polynomials by constructing natural scalar products on the spaces ℂ[S_{n}] and ℂ[S_{n}^{1}], proportional to each other in each fixed multidegree, such that the direct sum decompositions (9) and (10) given in the last section are orthogonal. For n = 2 this is equivalent to the classical orthogonality property just mentioned, and for n = 3 it is exactly analogous, because each summand in (9) and (10) has dimension ≤ 1, so that we again obtain a canonical orthogonal basis of the space of all polynomials on S_{n}^{1} . For n ≥ 4 the spaces P_{a}(d) in general have dimension greater than 1, and we could not find any natural orthogonal basis. However, it will turn out that there are two natural bases of P_{a}(d), one of them being the “monomial basis” constructed in the last section, that are dual to each other (at least up to scalar factors; the normalizations of both bases are not entirely canonical) with respect to the scalar product. This will be discussed in §5.

The definitions of both scalar products will be motivated by the case when d is an integer ≥ n, so that P_{a}(d) can be identified with (⊗H_{a_{i}}(ℝ^{d}))^{O(d)}. In the classical theory of harmonic polynomials, one defines a scalar product ( , )_{S^{d−1}} on ℂ[ℝ^{d}] by

(f, g)_{S^{d−1}} = \frac{1}{Vol(S^{d−1})} ∫_{S^{d−1}} f(x) \overline{g(x)} dx (f, g ∈ ℂ[ℝ^{d}]) ,

where dx denotes the standard volume form on the sphere, and shows that the spaces H_{a}(ℝ^{d}) and H_{a}(ℝ^{d}) are orthogonal with respect to this inner product if a ≠ b. Furthermore, the restriction map to S^{d−1} is injective on H_{a}(ℝ^{d}) and the space of all polynomial functions on S^{d−1} is the direct sum of these spaces. (In fact the space H_{a}(ℝ^{d})_{S^{d−1}} is just the eigenspace of the Laplacian operator Δ_{S^{d−1}} of the Riemannian manifold S^{d−1} with eigenvalue a(a + d − 2), and this orthogonal decomposition simply corresponds to the spectral decomposition of L^{2}(S^{d−1}) with respect to the Laplacian.)

For purposes of calculation it is convenient to relate the scalar product ( , )_{S^{d−1}} to a second scalar product ( , )_{ℝ^{d}} on ℂ[ℝ^{d}], defined by

(f, g)_{ℝ^{d}} = (2π)^{−d/2} ∫_{ℝ^{d}} f(x) \overline{g(x)} e^{−∥x∥^{2}/2} dx ,

where the normalizing factor (2π)^{−d/2} has been chosen so that (1, 1)_{ℝ^{d}} = 1. We can write the definitions of (f, g)_{S^{d−1}} and (f, g)_{ℝ^{d}} as E_{S^{d−1}}[f\overline{g}] and E_{ℝ^{d}}[f\overline{g}], where E_{S^{d−1}}[f] = (f, 1)_{S^{d−1}} and E_{ℝ^{d}}[f] = (f, 1)_{ℝ^{d}} denote the expectation values of f with respect to the probability measure Vol(S^{d−1})^{−1}dx and (2π)^{−d/2}e^{−∥x∥^{2}/2}dx on S^{d−1} and ℝ^{d}, respectively. For (the restrictions to S^{d−1} of) homogeneous polynomials these two measures are proportional: if f ∈ ℂ[ℝ^{d}]_{a}, then

E_{ℝ^{d}}[f] = (2π)^{−d/2} ∫_{0}^{∞} e^{−r^{2}/2} (∫_{S^{d−1}} f(rx) d(rx)) dr

= \frac{Vol(S^{d−1})}{(2π)^{d/2}} ∫_{0}^{∞} r^{a+d−1} e^{−r^{2}/2} dr · E_{S^{d−1}}[f]

= ε_{a}(d) E_{S^{d−1}}[f]  (13)

with

ε_{a}(d) = 2^{a/2} Γ(\frac{a + d}{2})/Γ(\frac{d}{2}) = d(d + 2) ⋯ (d + a − 2)  (a even).  (14)

(The definition of ε_{a}(d) for a odd does not matter because in that case f(x) is odd and both measures under consideration vanish trivially.)

The scalar products ( , )_{ℝ^{d}} and ( , )_{S^{d−1}} on ℂ[ℝ^{d}] extend in the obvious way to scalar products ( , )_{(ℝ^{d})^{n}} and ( , )_{(S^{d−1})^{n}} on ℂ[(ℝ^{d})^{n}] and induce scalar products ( , )_{d} and ( , )_{d}^{1} on ℂ[S_{n}] by the formulas (in which elements of (ℝ^{d})^{n} are thought of as n × d matrices)

(P, Q)_{d} = (P̃, Q̃)_{(ℝ^{d})^{n}} = (2π)^{−nd/2} ∫_{ℝ^{dn}}P (XX^{t}) \overline{Q(XX^{t})} e^{−tr(XX^{t})/2} dX ,  (15)
(P, Q)_{d}^{(1)} = (P̃, Q̃)_{(S^{d−1})^{n}} = \frac{1}{Vol(S^{d−1})^{n}} ∫_{(S^{d−1})^{n}}P (XX^{t}) \overline{Q(XX^{t})} dX  (16)

for P, Q ∈ ℂ[S_{n}], where P̃ = P ∘ β_{n}, Q̃ = Q ∘ β_{n} as usual. The properties reviewed above then immediately imply the corresponding properties for these new scalar products, namely, that the spaces P_{a}(d) = (⊗H_{a_{i}}(ℝ^{d}))^{O(d)} and P_{b}(d) = (⊗H_{b_{i}}(ℝ^{d}))^{O(d)} are orthogonal to each other if the multi-indices a and b are distinct, and that the two scalar products are proportional in each multidegree:

(P, Q)_{d} = ε_{a+b}(d) (P, Q)_{d}^{(1)} for P ∈ ℂ[S_{n}]_{a}, Q ∈ ℂ[S_{n}]_{b} . (17)

with ε_{a}(d) ∈ ℤ[d] defined by

ε_{a}(d) = ∏_{i=1}^{n} ε_{a_{i}}(d)  (a ∈ ℤ_{≥0}^{n}) .  (18)

(Again this definition is relevant only if a is divisible by 2, since, as one can see easily, polynomials whose multidegrees are not congruent modulo 2 are orthogonal with respect to both scalar products.) We now show that these definitions can be extended to arbitrary complex values of d, with the same properties.

Denote by S_{n}^{+} ⊂ S_{n} the space of positive definite symmetric n × n matrices and by S_{n}^{1,+} its intersection with the space S_{n}^{1} of n × n matrices with all diagonal coefficients equal to 1. The spaces S_{n} and S_{n}^{1} are isomorphic to the Euclidean spaces ℝ^{n(n+1)/2} and ℝ^{n(n−1)/2}, respectively, and have natural Lebesgue measures dT = ∏_{1≤i≤j≤n} dt_{ij} and d^{1}T = ∏_{1≤i<j≤n} dt_{ij} which we can restrict to their open subsets S_{n}^{+} and S_{n}^{1,+}. For ℜ(d) sufficiently large we define

(P, Q)_{d} = c_{n}(d) ∫_{S_{n}^{+}} e^{−tr(T)/2} P (T) \overline{Q(T)} det(T)^{(d−n−1)/2} dT  (19)

and

(P, Q)_{d}^{1} = c_{n}^{1}(d) ∫_{S_{n}^{1,+}} P (T) \overline{Q(T)} det(T)^{(d−n−1)/2} d^{1}T ,  (20)

where c_{n}(d) and c_{n}^{1}(d) are normalizing constants defined by

2^{nd/2} c_{n}(d) = Γ(\frac{d}{2})^{−n} c_{n}^{1}(d) = π^{−n(n−1)/4} ∏_{i=0}^{n−1} Γ(\frac{d − i}{2})^{−1} .  (21)

Theorem 2. (a) The integrals in (19) and (20) converge absolutely for d ∈ ℂ with ℜ(d) > n−1 and their right-hand sides agree with (15) and (16) if d is an integer ≥ n.

(b) The values of (P, Q)_{d} or (P, Q)_{d}^{1} for P and Q multihomogeneous are related as in (17) for all complex values of d, with ε_{a}(d) defined by equations (14) and (18).

(c) For fixed P, Q ∈ ℂ[S_{n}] and variable d, the scalar product (P, Q)_{d} is a polynomial in d, belonging to ℤ[d] if P and Q have integral coefficients, and (1, 1)_{d} = 1.

(d) The spaces P_{a}(d) and P_{b}(d) with distinct multi-indices a and b are orthogonal.

Remark 1. Parts (b) and (c) give the holomorphic continuation of ( , )_{d} and the meromorphic continuation of ( , )_{d}^{1} to all complex values of d, with the poles of the latter occurring only for d ∈ {0, −2, −4, … }. These are the same exceptional values as in Theorem 1.

Remark 2. Part (d) (in which we do not have to specify which scalar product is meant, by part (b)) implies that both the direct sum decomposition ℂ[S_{n}] = ⨁_{a} P_{a}(d) ⊗ ℂ[t_{11}, …, t_{nn}] given by the corollary to Theorem 1 and the direct sum decomposition of ℂ[S_{n}]_{a} given in (9) are orthogonal with respect to either scalar product, and that the direct sum decomposition of ℂ[S_{n}^{1}] given in (10) is orthogonal with respect to the scalar product ( , )_{d}^{1} on ℂ[S_{n}^{1}].

Remark 3. If d is real and > n − 1, then the convergent integral representation (19) shows that the scalar product ( , )_{d} on ℂ[S_{n}] is positive definite and hence non-degenerate. For other values of d, ( , )_{d} may be degenerate. We will show in Theorem 11 that this can happen only if d is an integer. Numerical examples can be found at the end of §5 and in Tables 2 and 3 at the end of the paper.

Remark 4. Under the correspondence P ↔ p between functions in P ∈ ℂ[S_{n}]_{a} and p ∈ ℂ[S_{n}^{1,+}] given in (6), we have D_{i}(P) ↔ (𝔻_{i}+a_{i}(a_{i}+d−2))p for some second order differential operators 𝔻_{i} (depending on d but not on a) on S_{n}^{1,+}. (Compare (8) for the case n = 3, i = 1.) The spaces P_{a}(d) therefore correspond to spaces of simultaneous eigenfunctions of the operators 𝔻_{i}, with eigenvalues a_{i}(a_{i} + d − 2). If d is real and > n − 1, then we can define a Hilbert space L^{2}(S_{n}^{1,+}) using the positive definite scalar product (20). Since S_{n}^{1,+} is compact and the restrictions to S_{n}^{1} of polynomials on S_{n} separate points, these restrictions are dense in L^{2}(S_{n}^{1,+}) by the Stone-Weierstrass theorem. It follows that L^{2}(S_{n}^{1,+}) has an orthogonal Hilbert space direct sum decomposition as ⨁_{a} P_{a}(d)|_{S_{n}^{1,+}} and that this is simply the eigenspace decomposition of the collection of commuting self-adjoint operators 𝔻_{i}.

Proof of Theorem 2. We begin by proving (b), the relationship between the two scalar products (P, Q)_{d} and (P, Q)_{d}^{1} when P and Q are multihomogeneous, after which we can work only with (P, Q)_{d}, which is easier. We can also simplify by observing that (P, Q)_{d} and (P, Q)_{d}^{1} can be written as E_{d}[P Q̄] and E_{d}^{1}[P Q̄], respectively, where E_{d} and E_{d}^{1} are the maps from ℂ[S_{n}] to ℂ defined by E_{d}[P] = (P, 1)_{d} and E_{d}^{1}[P] = (P, 1)_{d}^{1}. This is more convenient since these functions depend on only one rather than on two arguments. The statement of (b) then becomes that E_{d}[P] = ε_{a}(d) E_{d}^{1}[P] for P ∈ ℂ[S_{n}]_{a}. To prove it, we observe that every T ∈ S_{n}^{+} can be written uniquely as λT_{1}λ where λ is a diagonal matrix with positive entries and T_{1} ∈ S_{n}^{1}, the values of λ and T being given by t_{ii} = λ_{i}^{2} and t_{ij} = λ_{i}λ_{j}τ_{ij} for i ≠ j, where τ_{ij} are the entries of T_{1}. We then have

dT = ∏_{i=1}^{n}(2λ_{i} dλ_{i}) · ∏_{i<j}(λ_{i}λ_{j}dτ_{ij}) = (2λ_{1} ⋯ λ_{n})^{n} dT_{1} dλ ,

and of course det(T) = (λ_{1} ⋯ λ_{n})^{2} det(T_{1}), so (4) and the definitions of E_{d} and E_{d}^{1} give

E_{d}[P] = 2^{n} c_{n}(d) ∫_{ℝ_{+}^{n}} ∫_{S_{n}^{1,+}} e^{−∥λ∥^{2}/2} λ^{a} P (T_{1}) (λ_{1} ⋯ λ_{n})^{d−1} det(T_{1})^{(d−n−1)/2} dT_{1} dλ

= E_{d}^{1}[P] · \frac{c_{n}(d)}{c_{n}^{1}(d)} ∏_{i=1}^{n}(2 ∫_{0}^{∞} e^{−λ^{2}/2} λ^{d+a_{i}−1} dλ) = ε_{a}(d) E_{d}^{1}[P]

for P ∈ ℂ[S_{n}]_{a}, where the last equality follows from by (18), (14) and the first of equations (21) because the expression in parentheses equals 2^{d/2}Γ(d/2) ε_{a_{i}}(d) for a_{i} even and because both E_{d}[P] and E_{d}^{1}[P] vanish if a_{i} is odd (as one sees by replacing T by λT λ, where λ is the diagonal matrix with λ_{i} = −1 and λ_{j} = 1 for j ≠ i).

We next show the equality of the right-hand sides of (19) and (15) when d is an integer ≥ n. (The equality of the right-hand sides of (20) and (16) then also follows since we have proved the proportionality result (17) for both definitions of the scalar products.) We must show that

c_{n}(d) ∫_{S_{n}^{+}} F (T) det(T)^{(d−n−1)/2} dT = (2π)^{−nd/2} ∫_{(ℝ^{d})^{n}} F (XX^{t}) dX .  (22)

for rapidly decreasing functions F : S_{n}^{+} → ℂ. (Apply this to F (T) = P (T) \overline{Q(T)} e^{−tr(T)/2}.) To do this, we will decompose the Lebesgue measure on ℝ^{dn} into a T -part and another part. We first decompose X as SV where S is a lower triangular n × n matrix with positive diagonal entries and V an n × d matrix with orthonormal rows, i.e., we write each x_{i} (1 ≤ i ≤ n) as ∑_{1≤j≤i} s_{ij} v_{j} with s_{ii} > 0 and v_{1}, …, v_{n} orthogonal unit vectors in ℝ^{d} (Gram-Schmidt orthonormalization process). The numbers s_{ij} and the components of the vectors v_{j} can be taken as local coordinates, so we can decompose dX into an S-part and a V -part. The vector v_{1} belongs to the unit sphere in ℝ^{d}, so for the vector x_{1} = s_{11}v_{1} we get dx_{1} = s_{11}^{d−1} ds_{11} dμ_{d−1}, where dμ_{d−1} denotes the appropriately normalized standard measure on the sphere S^{d−1}. Once we have fixed v_{1}, then the vector v_{2} belongs to a (d − 2)-dimensional sphere and for the vector x_{2} = s_{21}v_{1} + s_{22}v_{2} we get dx_{2} = s_{22}^{d−2} ds_{21}ds_{22} dμ_{d−2}. Continuing the process, we get

dX = ∏_{1≤i≤n} s_{ii}^{d−i} ∏_{1≤j≤i≤n} ds_{ij} dV ,

where dV := dμ_{d−1} dμ_{d−2} ⋯ dμ_{d−n} is the measure on the V -coordinates. By definition, we have T = XX^{t} = SV V^{t}S^{t} = SS^{t}. This gives

dT = ∏_{i≤j} dt_{ij} = 2^{n} ∏_{i=1}^{n} s_{ii}^{n−i+1} ∏_{i≥j} ds_{ij} .  (23)

(We have t_{ii} = s_{ii}^{2} + ⋯ and t_{ij} = s_{ii}s_{ji} + ⋯ for j > i, where ⋯ denotes terms of higher index if we order the pairs (i, j) lexicographically.) Comparing the last two formulas and noting that ∏_{i=1}^{n} s_{ii} = det(T)^{1/2}, we get

dX = 2^{−n} det(T)^{\frac{d−n−1}{2}} ∏_{i≤j} dT dV ,

and since V runs over ∏_{i=0}^{n−1} S^{d−i−1} this implies (22), with c_{n}(d) given by

c_{n}(d) = 2^{−n} (2π)^{−nd/2} ∏_{i=0}^{n−1} Vol(S^{d−i−1}) = 2^{−nd/2} π^{−n(n−1)/4} ∏_{i=0}^{n−1} Γ(\frac{d − i}{2})^{−1} ,

in agreement with the formula in (21).

We now see that essentially the same calculation works also for non-integral values of d, because the decomposition T = SS^{t} did not involve the V -part of the variable X ∈ ℝ^{nd} but only the n × n matrix S. The maps S ↦ SS^{t} defines a bijection between the space 𝔏_{n}^{+} of lower triangular n × n matrices S with positive diagonal entries and the space S_{n} of positive definite symmetric n×n matrices. The space 𝔏_{n}^{+} is an open subset of a Euclidean space with coordinates s_{ij} (1 ≤ j ≤ i ≤ n) and natural Lebesgue measure dS = ∏_{j≥i} ds_{ij}, so by (23) we find

E_{d}[P] = 2^{n} c_{n}(d) ∫_{𝔏_{n}^{+}} e^{−tr(SS^{t})/2} P̂(S) ∏_{i=1}^{n} s_{ii}^{d−i} dS =: E_{d}^{𝔏}[P̂]  (24)

for any P ∈ ℂ[S_{n}], where P̂ ∈ ℂ[𝔏_{n}]^{ev} (the space of polynomials on lower triangular n × n matrices that are invariant under S ↦ Sλ for any diagonal matrix λ with all entries ±1) is defined by P̂(S) = P (SS^{t}). (The formula for P̂(S) is the same as the formula for P̃(S) in the special case n = d if we consider 𝔏_{n} as a subspace of the space (ℝ^{n})^{n} of all n × n matrices, but the roles of X and S are quite different and we prefer to use a different notation.) This formula makes it clear that the integral converges for all polynomials if (and only if) d is a complex number with real part > n − 1, since for convergence in (24) we need d − i > −1 for all 1 ≤ i ≤ n. This completes the proof of part (a) of the theorem, but part (c) now also follows immediately, since we have the explicit formula

E_{d}^{𝔏}[∏_{j≤i} s_{ij}^{m_{ij}}] = 2^{n}c_{n}(d) ∏_{1≤j<i≤n}(∫_{−∞}^{∞} s^{m_{ij}}e^{−s^{2}/2} ds) · ∏_{1≤i≤n}(∫_{0}^{∞} s^{m_{ii}+d−i}e^{−s^{2}/2} ds)

= \{ \begin{matrix}∏_{1≤j<i≤n}(m_{ij} − 1)!! · ∏_{1≤i≤n} ε_{m_{ii}}(d − i + 1) & if all m_{ij} are even \\ 0 & otherwise, \end{matrix}  (25)

where (m − 1)!! is defined as usual as 1 for m = 0 and as 1 × 3 × ⋯ × (m − 1) for m > 0 odd. (Here one has to note that the integral over ds_{ij} for j < i vanishes if m_{ij} is odd because we are integrating an odd function over all of ℝ, and that if all of the m_{ij} with j ≠ i are even then the diagonal entries are always even because for a monomial ∏ s_{ij}^{m_{ij}} in ℂ[𝔏_{n}]^{ev} the sum of the exponents m_{ij} for any fixed value of j is even. We are therefore in the case where the second formula in equation (14) can be applied.) It thus follows that E_{d}[P] = E_{d}^{𝔏}[P̂] ∈ ℤ[d] for any polynomial P ∈ ℤ[S_{n}], and that E_{d}[1] = 1, proving (c).

Finally, we observe that the orthogonality statement (d), the most important part of the theorem, now follows formally from the other parts, since we already know that it is true when d is an integer ≥ n and a polynomial which vanishes for infinitely many values of its argument must be identically zero. A more direct proof for arbitrary d will follow from results given in the next section, where we will show that

(a_{i} − b_{i}) (P, Q)_{d} = (D_{i}P, Q)_{d} − (P, D_{i}Q)_{d}  (26)

for every P ∈ ℂ[S_{n}]_{a}, Q ∈ ℂ[S_{n}]_{b} and every 1 ≤ i ≤ n, and hence (a_{i} − b_{i})(P, Q)_{d} = 0 if P, Q ∈ Ker(D_{i}). □

We now discuss the actual calculation of the scalar products. In the case when d ≥ n is an integer, this is easy because the expression E_{ℝ^{d}}[f] can be calculated trivially for monomials. (For d = 1 one has E_{ℝ}[x^{m}] = (m − 1)!! for m even and E_{ℝ}[x^{m}] = 0 for m odd, and the values for ℝ^{d} or (ℝ^{d})^{n} are obtained from this by multiplicativity.) For d complex the integral (19), although it is convergent if d is large enough, is useless for computations since it involves a complex power of a polynomial with many terms. We mention three^{3} quite different effective methods—each of which could be used as an alternative definition instead of the integral (19)—for calculating the scalar products (P, Q)_{d} explicitly as polynomials in d.

(i) The first way is to use the formula (24) together with the explicit evaluation (25). This can be programmed very easily.

(ii) The second method is to use rules that will be given in the next section (Proposition 1) to write the value of E_{d} on any polynomial of the form t_{ij}P in terms of its values on P and derivatives of P. This gives E_{d} for all monomials inductively starting from the value E_{d}[1] = 1.

(iii) The third, and in some sense most explicit, method comes from the following theorem, which gives a generating function for all monomials P (T) = T^{ν ν ν}. To state it, we introduce the notation

ννν!!! = ∏_{1≤i≤n} ν_{ii}!! · ∏_{1≤i<j≤n} ν_{ij}!  (27)

for ννν = (ν_{ij})_{1≤i, j≤n} ∈ N_{n}, where ν!! is defined as 2^{ν/2}(ν/2)! for ν even. Then we have:

Theorem 3. Let Y be a symmetric matrix with complex entries with 1 − Y \overline{Y^{t}} > 0. Then

∑_{ν ν ν∈N_{n}} E_{d}[T^{ν ν ν}] \frac{Y^{ν ν ν}}{ννν!!!} = det(1 − Y )^{−d/2} = exp(\frac{d}{2} ∑_{r=1}^{∞} \frac{tr(Y^{r})}{r}) .  (28)

Proof. Both series in (28) converge exponentially under the condition stated, since then all eigenvalues of Y are less than 1 in absolute value. (This is actually not very important for us, since in applications of the theorem Y will always be a formal variable.) By analytic continuation, we can assume that Y is real. The second equality in (28) is standard, so we need only prove the first. The basic calculation is

∑_{ν ν ν∈N} \frac{A^{ν ν ν}}{ννν!!!} = ∏_{1≤i≤n}(∑_{ν=0}^{∞} \frac{a_{ii}^{ν}}{2^{ν}ν!}) · ∏_{1≤i<j≤n}(∑_{ν=0}^{∞} \frac{a_{ij}^{ν}}{ν!} ) = exp(\frac{1}{2} ∑_{1≤i,j≤n} a_{ij}) ,

valid for any symmetric n×n matrix A. Applying it to the matrix with entries a_{ij} = t_{ij}y_{ij} gives

∑_{ν ν ν∈N} \frac{T^{ν ν ν}Y^{ν ν ν}}{ννν!!!} = e^{tr(T Y)/2}  (29)

for any T, Y ∈ S_{n} and hence

∑_{ν ν ν∈N_{n}} E_{d}[T^{ν ν ν}] \frac{Y^{ν ν ν}}{ννν!!!} = c_{n}(d) ∫_{S_{n}^{+}} e^{−tr(T (1−Y))/2} det(T)^{(d−n−1)/2} dT

for sufficiently small Y ∈ S_{n}. We can write 1 − Y = U^{2} for some invertible n × n matrix U ∈ S_{n}. Then tr(T (1 − Y)) = tr(UT U). The change of variables from T to UT U is a bijection from S_{n}^{+} to itself and sends det(T) to det(U)^{2} det(T) and dT to det(U)^{n+1}dT, so the value of the expression on the right is det(U)^{−d} times its value for U = 1, which equals 1 by the definition of the normalizing constant c_{n}(d). □

3Another effective method uses the Corollary to Proposition 2 in §4 below. All values given in the tables at the end of the paper were computed independently from at least two of these four formulas.

We remark that a related, but somewhat simpler proof of Theorem 3 can be given in the case when d ≥ n is an integer (which suffices to prove the general case since (28) is equivalent to a collection of polynomial identities in d). Again using the identity (29), we find in this case

∑_{ν ν ν∈N_{n}} E_{d}[T^{ν ν ν}] \frac{Y^{ν ν ν}}{ννν!!!} = (2π)^{−nd/2} ∫_{(ℝ^{d})^{n}} e^{−tr(XX^{t}(1−Y))/2} dX

= (2π)^{−nd/2} ∫_{(ℝ^{n})^{d}} e^{−tr(X(1−Y)X^{t})/2} dX (X → X^{t})

= ((2π)^{−n/2} ∫_{ℝ^{n}} e^{−x(1−Y)x^{t}/2} dx)^{d} = det(1 − Y)^{−d/2} ,

where we have used the standard identity π^{−n/2} ∫_{ℝ^{n}} e^{−xAx^{t}}dx = (det A)^{−1/2} for A ∈ S_{n}^{+}. As a simple example of the application of the theorem, we have

∑_{m=0}^{∞} E_{d}[t_{ii}^{m}] \frac{y^{m}}{2^{m}m!} = (1 − y)^{−d/2} = ∑_{m=0}^{∞} \frac{ε_{2m}(d)}{2^{m}m!} y^{m} .

Of course here we knew the answer already since E_{d}[t_{ii}^{m}] = ε_{2m}(d) E_{d}^{1}[t_{ii}^{m}] = ε_{2m}(d) E_{d}^{1}[1] = ε_{2m}(d). As a less trivial example, we take Y = y e_{ij} (e_{ij} as in §1) for i ≠ j to get

∑_{m=0}^{∞} E_{d}[t_{ij}^{m}] \frac{y^{m}}{m!} = det(1 − y e_{ij})^{−d/2} = (1 − y^{2})^{−d/2}

and hence

E_{d}[t_{ij}^{m}] = \{ \begin{matrix}(m − 1)!! · 2^{m/2} (d/2)_{m/2} & if m is even, \\ 0 & if m is odd. \end{matrix}  (30)

§4. The Lie algebra of mixed Laplacians

Let us return for the moment to the case that d is an integer ≥ n and that our polynomials on S_{n} correspond via P ↦ P̃ to the O(d)-invariant polynomials on (ℝ^{d})^{n}. Then as well as the “pure” Laplace operators

Δ_{i} = Δ_{ii} = (\frac{∂}{∂x_{i}} )(\frac{∂}{∂x_{i}} )^{t} = ∑_{α=1}^{d} \frac{∂^{2}}{∂x_{iα}^{2}} (1 ≤ i ≤ n)

on ℂ[(ℝ^{d})^{n}] that were used to define P_{a}(d), we also have the “mixed” Laplace operators

Δ_{ij} = (\frac{∂}{∂x_{i}} )(\frac{∂}{∂x_{j}} )^{t} = ∑_{α=1}^{d} \frac{∂^{2}}{∂x_{iα} ∂x_{jα}} (1 ≤ i, j ≤ n) .

The same calculation as for Δ_{i} shows that these correspond under the map (1) to the differential operators

D_{ij} = D_{ij}^{(d)} = d ∂_{ij} + ∑_{k,l=1}^{n} t_{kl} ∂_{ik}∂_{jl} .  (31)

The D_{ij}, which can now be defined for this equation also for d < n or d ∉ ℤ, commute with one other and in particular with the pure Laplacians D_{i} = D_{ii}, so they preserve the condition of being harmonic, i.e. D_{ij} (i ≠ j) maps P_{a}(d) to P_{a−e_{i}−e_{j}}(d), where e_{i} is the vector with 1 in the ith place and 0’s elsewhere. These “intertwining” operators will turn out to be very useful.

In exactly the same way, as well as the operators E_{i} : ℂ[S_{n}]_{a} → ℂ[S_{n}]_{a} corresponding to the Euler operators ∑_{α} x_{i,α}∂/∂x_{i,α} (Remark following the Proposition-Definition in §1) we also have maps E_{ij} : ℂ[S_{n}]_{a} → ℂ[S_{n}]_{a+e_{i}−e_{j}} for i ≠ j, given explicitly by E_{ij} = ∑_{k=1}^{n} t_{ik} ∂_{jk}, which correspond when d is integral and T = XX^{t} to the “mixed Euler operators” ∑_{α} x_{i,α}∂/∂x_{j,α} (because these are again O(d)-invariant). Finally, we have operators F_{ij} : ℂ[S_{n}]_{a} → ℂ[S_{n}]_{a+e_{i}+e_{j}} for all i, j = 1, …, n given by multiplication with t_{ij}. In this section we will show that the vector space spanned by all of these operators and by the identity is closed under commutators and also under adjoints with respect to the scalar product defined in the previous section.

To check the Lie property involves computing a large number of different commutators. We have six kinds of commutators ([D, D], [D, E], [D, F], [E, E], [E, F] and [F, F]) and for each one a number of different special cases depending on which indices are distinct and which are equal. For instance, to give the commutators of the operators D_{ij} with F_{kl} we have to consider seven different cases, obtaining after somewhat tedious computations the seven formulas

[D_{ii}, F_{ii}] = 4E_{i} + 2d, [D_{ii}, F_{ik}] = 2E_{ki}, [D_{ii}, F_{kl}] = 0,
[D_{ij}, F_{ij}] = E_{i} + E_{j} + d, [D_{ij}, F_{ii}] = 2E_{ij}, [D_{ij}, F_{ik}] = E_{kj}, [D_{ij}, F_{kl}] = 0

for i ≠ j and k, l ≠ i, j. If we modify the definition of E_{ij} for i = j by setting E_{ii} = E_{i} + d/2 rather than by the more natural-seeming formula E_{ii} = E_{i}, then the scalar terms “2d” and “d” in these equations drop out and the seven cases can be written in a uniform way. Moreover, somewhat surprisingly, the corresponding simplification also occurs in all the other types of commutators if we make the same substitution; in other words, the vector space spanned by the operators D_{ij}, E_{ij} and F_{ij} is already closed under commutators, without having to include also the identity operator. More precisely, we have:

Theorem 4. For fixed n ∈ ℤ_{≥0} and d ∈ ℂ, the vector space 𝔤 ⊂ End(ℂ[S_{n}]) spanned by

D_{ij} = d∂_{ij} + ∑_{k,l=1}^{n} t_{kl}∂_{ik}∂_{jl} , E_{ij} = \frac{d}{2} δ_{ij} + ∑_{k=1}^{n} t_{ik}∂_{jk} , F_{ij} = t_{ij} (1 ≤ i, j ≤ n) (32)

is a Lie subalgebra, with commutators given by

[D_{ij}, D_{kl}] = 0, [D_{ij}, E_{kl}] = δ_{ik}D_{jl} + δ_{jk}D_{il}, [D_{ij}, F_{kl}] = δ_{ik}E_{lj} + δ_{il}E_{kj} + δ_{jk}E_{li} + δ_{jl}E_{ki}, [E_{ij}, E_{kl}] = δ_{jk}E_{il} − δ_{il}E_{kj}, [E_{ij}, F_{kl}] = δ_{jk}F_{il} + δ_{jl}F_{ik}, [F_{ij}, F_{kl}] = 0 (1 ≤ i, j, k, l ≤ n) .

Proof. As already indicated, this is just a straightforward but lengthy computation, with many cases to be checked. We give the proof of the second commutation relation as an example. From the definitions we have

[D_{ij}, E_{kl}] = d ∑_{m}[∂_{ij}, t_{km}∂_{lm}] + ∑_{r,s,m}[t_{rs}∂_{ir}∂_{js}, t_{km}∂_{lm}] ,

where all indices run from 1 to n. The first term is equal to d ∑_{m}(δ_{ik}δ_{jm} + δ_{im}δ_{jk})∂_{lm} = d (δ_{ik}∂_{jl} + δ_{jk}∂_{il}), and the second to

∑_{r, s, m}[(δ_{jk}δ_{sm} + δ_{jm}δ_{ks})t_{rs}∂_{ir}∂_{lm} + (δ_{ik}δ_{rm} + δ_{im}δ_{rk})t_{rs}∂_{js}∂_{lm} − (δ_{lr}δ_{ms} + δ_{ls}δ_{mr})t_{km}∂_{ir}∂_{js}]

= δ_{jk} ∑_{r, s} t_{rs}∂_{ir}∂_{ls} + (1 − 1) ∑_{r} t_{kr}∂_{ir}∂_{jl} + δ_{ik} ∑_{r, s} t_{rs}∂_{jr}∂_{ls} + (1 − 1) ∑_{s} t_{ks}∂_{js}∂_{il} .

Adding these gives the desired result [D_{ij}, E_{kl}] = δ_{ik}D_{jl} + δ_{jk}D_{il}. The other statements in the theorem can be checked in the same way. Alternatively, since each of the commutation relations among the generators of 𝔤 is polynomial in d and therefore need only be checked for values of d ∈ ℤ_{≥n}, we can work instead with the mixed Laplace and Euler operators and the operations of multiplying by (x_{i}, x_{j}) on the space of polynomials in (ℝ^{d})^{n} and check the commutation relations there; then the calculations are different but the final result is the same. □

Observe that the operators D_{ij} and F_{ij} are symmetric in the two indices i and j, whereas E_{ij}

is not, so the dimension of the Lie algebra 𝔤 is equal to \frac{n(n + 1)}{2} + n^{2} + \frac{n(n + 1)}{2} = 2n^{2} + n. This is the same as the dimension of the simple Lie algebra

𝔰𝔭(2n, ℝ) = \{X ∈ M_{2n}(ℝ) | XJ_{n} = −J_{n}X^{t}\} , J_{n} = ( \begin{matrix}0 & −1_{n} \\ 1_{n} & 0 \end{matrix} ) ,

and indeed (working over ℝ) we have an isomorphism 𝔤 ≅ 𝔰𝔭(2n, ℝ), as we can see explicitly by making the assignments

D_{ij} ↦ ( \begin{matrix}0 & 0 \\ −e_{ij} − e_{ji} & 0 \end{matrix} ) , E_{ij} ↦ ( \begin{matrix}e_{ij} & 0 \\ 0 & −e_{ji} \end{matrix} ) , F_{ij} ↦ ( \begin{matrix}0 & e_{ij} + e_{ji} \\ 0 & 0 \end{matrix} )

(where e_{ij} (1 ≤ i, j ≤ n) is the elementary n × n matrix with 1 in the (i, j)th place and zeros elsewhere) and comparing the commutators in 𝔤 as given in the proposition with those in the Lie algebra 𝔰𝔭(2n). A better understanding of why 𝔰𝔭(2n) occurs comes from the discussion given in §1 about the connection between higher spherical polynomials and Siegel modular forms. The Lie group Sp(2n, ℝ) acts on the space of holomorphic functions F : ℌ_{n} → ℂ by (F |g)(Z) = det(CZ + D)^{−d/2}F ((AZ + B)(CZ + D)^{−1}) for g = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ G. (This is the action used to define Siegel modular forms of weight d/2.) The action of G induces an action F ↦ F |X of its Lie algebra 𝔰𝔭(2n, ℝ) via F |e^{εX} = F + (F |X)ε + O(ε^{2}), and from the calculation (setting

ε^{2} = 0 and X = ( \begin{matrix}A & B \\ C & D \end{matrix} ))

F |( \begin{matrix}1+εA & εB \\ εC & 1+εD \end{matrix} )(Z) ≡ det(1 + ε(CZ + D))^{−d/2} F ((Z + ε(AZ + B))(1 + ε(CZ + D))^{−1})

≡ (1 − ε \frac{d}{2} tr(CZ + D)) F (Z + ε(AZ + B − Z(CZ + D)))

we see that this action is given explicitly by

F |( \begin{matrix}A & B \\ C & D \end{matrix} )(Z) = − \frac{d}{2} tr(CZ + D) F (Z) + ∑_{i, j=1}^{n} (AZ + B − Z(CZ + D))_{ij} ∂_{ij}^{∗}F (Z) ,

where ∂_{ij}^{∗} = \frac{1}{2}(1 + δ_{ij})∂/∂z_{ij} . Using the above isomorphism 𝔤 ≅ 𝔰𝔭(2n, ℝ), we then find

D_{ij} ↦ dz_{ij} + ∑_{k,l=1}^{n} z_{ik}z_{jl} ∂_{kl}^{∗} , E_{ij} ↦ \frac{d}{2} δ_{ij} + ∑_{k=1}^{n} z_{ik} ∂_{jk}^{∗} , F_{ij} ↦ ∂_{ij}^{∗}

and this agrees with the definition of the generators in 𝔤 if we remember from the discussion in §1 that the variables T (argument of the higher spherical polynomials) and Z (argument of the Siegel modular forms) are dual variables with respect to the Fourier expansion, so that t_{ij} and ∂_{ij} correspond to ∂_{ij}^{∗} and z_{ij}. In fact, we can use this correspondence to see directly that the generators of 𝔤 correspond to the standard generators of 𝔰𝔭(2n), thus obtaining yet a third proof of the commutation relations in Theorem 2. This is certainly related to the Weil representation ([15], [13]), but we will not discuss this any further here because the identification of 𝔤 with 𝔰𝔭(2n, ℝ) will not play any role in this paper.

We now relate the operators in 𝔤 to the scalar products that were defined in §3. Since each of the generators in (32) is homogeneous, in the sense that it sends ℂ[S_{n}]_{a} to ℂ[S_{n}]_{a+δδδ} for some fixed δδδ ∈ ℤ^{n}, and since the two scalar products ( , )_{d} and ( , )_{d}^{1} are proportional in each multidegree, we only have to give the formulas for ( , )_{d}, where they are simpler. Recall from §3 that (P, Q)_{d} = E_{d}[P Q] for the linear map E_{d} : ℂ[S_{n}] → ℂ defined by E_{d}[P] = E_{(ℝ^{d})^{n}}[P̃].

Proposition 1. For arbirary d ∈ ℂ, i, j ∈ {1, …, n} and P ∈ ℂ[S_{n}] we have

E_{d}[F_{ij}P] = E_{d}[(E_{ij} + \frac{d}{2} δ_{ij})P ] = E_{d}[(D_{ij} + d δ_{ij})P] .  (33)

Proof. Since each expression in (33) is a polynomial in d for any P, we may assume that d ∈ ℤ_{≥n} and that ( , )_{d} is defined by (15). Note first that the operator E_{ℝ}[f] = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} f(x) e^{−x^{2}/2} dx (the special case n = d = 1 of E_{d}) satisfies the relation E_{ℝ}[xf] = E_{ℝ}[f′] for any f ∈ ℂ[x], as one can see either from the explicit formulas E_{ℝ}[x^{2m}] = (2m − 1)!! and E_{ℝ}[x^{2m−1}] = 0 or, more naturally, by using the identity

E_{ℝ}[f′] − E_{ℝ}[xf] = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} d(e^{−x^{2}/2} f(x)) = 0 ,

i.e., by integration by parts. This immediately gives E_{ℝ^{d}}[x_{α}f] = E_{ℝ^{d}}[∂f/∂x_{α}] (α = 1, …, d). for any f ∈ ℂ[ℝ^{d}] and hence

E_{(ℝ^{d})^{n}}[x_{iα}f] = E_{(ℝ^{d})^{n}}[\frac{∂f}{∂x_{iα}} ]  (i = 1, …, n, α = 1, …, d) .  (34)

for any f ∈ ℂ[(ℝ^{d})^{n}]. These formulas allow us to compute E_{ℝ}, E_{ℝ^{d}} and E_{(ℝ^{d})^{n}} on all monomials (and hence on all polynomials) by induction on the degree.

Using (34) twice, we find, for all F ∈ ℂ[(ℝ^{d})^{n}] and all 1 ≤ i, j ≤ n, 1 ≤ α ≤ d,

E_{(ℝ^{d})^{n}}[x_{iα}x_{jα}F (X)] = E_{(ℝ^{d})^{n}}[\frac{∂}{∂x_{jα}} (x_{iα}F )]

= E_{(ℝ^{d})^{n}}[δ_{ij}F + x_{iα} \frac{∂F}{∂x_{jα}} ] = E_{(ℝ^{d})^{n}}[δ_{ij}F + \frac{∂^{2}F}{∂x_{iα}∂x_{jα}} ]

and hence, summing over α,

E_{(ℝ^{d})^{n}}[ (x_{i}, x_{j}) F ] = E_{(ℝ^{d})^{n}}[ d δ_{ij}F + ∑_{α=1}^{d} x_{iα} \frac{∂F}{∂x_{jα}} ] = E_{(ℝ^{d})^{n}}[ d δ_{ij} F + Δ_{ij}F ] . (35)

Applying this to F = P̃ with P ∈ ℂ[S_{n}], we obtain (33) for all d ∈ ℤ_{≥n} and hence for all d. □

Notice that either of the equalities E_{d}[t_{ij}P] = E_{d}[(E_{ij} + \frac{d}{2}δ_{ij})P] or E_{d}[t_{ij}P] = E_{d}[(D_{ij} +dδ_{ij})P] in (33), together with the normalization E_{d}[1] = 1, defines E_{d} uniquely, because the argument of E_{d} on the right has lower degree than the one on the left. Proposition 1 therefore gives us a new definition (to be added to the three previous definitions (19), (24)/(25) and (28)) of the functional E_{d} : ℂ[S_{n}] → ℂ and hence of the scalar product ( , )_{d} for arbitrary d ∈ ℂ. Notice also that the first equality in (33) for that i = j gives the useful identity

E_{d}[t_{ii}P] = (d + a_{i}) E_{d}[P]  for P ∈ ℂ[S_{n}]_{a} .  (36)

This also follows from (17), since E_{d}[t_{ii}P] = ε_{a+2e_{i}}(d)E_{d}^{1}[t_{ii}P] = ε_{a+2e_{i}}(d)E_{d}^{1}[P] = (d + a_{i})E_{d}[P].

Proposition 1 describes the relation between the Lie algebra 𝔤 and the functional P ↦ (P, 1)_{d}. The following more precise result describes its relation to the whole scalar product.

Proposition 2. The Lie algebra 𝔤 is equal to its own adjoint with respect to the action on the space ℂ[S_{n}] with the inner product (P, Q)_{d}. Explicitly, the adjoint operators of the standard basis elements of 𝔤 are given by

D_{ij}^{∗} = D_{ij} − E_{ij} − E_{ji} + F_{ij} ,  E_{ij}^{∗} = −E_{ij} + F_{ij} ,  F_{ij}^{∗} = F_{ij} .  (37)

Proof. The third equality in (37) is trivial since (t_{ij}P) Q = P (t_{ij}Q) for any P and Q; notice that it does not matter whether we write “Q” or “\overline{Q}” since the integrations defining the scalar products are taken over real-valued variables. The second equality (which includes the third since E_{ij} + E_{ij}^{∗} is self-adjoint) follows from the first equality in (33) because E_{ij} − \frac{d}{2}δ_{ij} is a derivation. To prove the first equality, we assume once again that d ∈ ℤ_{≥n} and that the scalar product is defined by (15). For F and G in ℂ[(ℝ^{d})^{n}] we have

Δ_{ij}(F G) = Δ_{ij}(F) G + F Δ_{ij}(G) + ∑_{α=1}^{d}(\frac{∂F}{∂x_{iα}} \frac{∂G}{∂x_{jα}} + \frac{∂F}{∂x_{jα}} \frac{∂G}{∂x_{iα}} )

and hence, denoting by ≡ congruence modulo the kernel of E_{d} and using (34) and (35),

F Δ_{ij}(G) − Δ_{ij}(F) G = −Δ_{ij}(F G) + ∑_{α=1}^{d}[\frac{∂}{∂x_{iα}} (F \frac{∂G}{∂x_{jα}} ) + \frac{∂}{∂x_{jα}} (F \frac{∂G}{∂x_{iα}} )]

≡ − ((x_{i}, x_{j}) − d δ_{ij}) F G + F ∑_{α=1}^{d}(x_{iα} \frac{∂G}{∂x_{jα}} + x_{jα} \frac{∂G}{∂x_{iα}} )

= F (E_{ij} + E_{ji} − F_{ij}) G .

The identity now follows by applying E_{(ℝ^{d})^{n}} to both sides and replacing F and G by P̃ and Q̃ with P, Q ∈ ℂ[S_{n}]. □

Observe that equation (26), which was used in the last section to give a second and more direct proof of the orthogonality of P_{a}(d) and P_{b}(d) for a ≠ b, follows from (36) and the special case i = j of the first identity in (37).

As a further consequence of Proposition 2, we get the following nice formula for the scalar product of §3 with respect to the monomial basis of P(d).

Corollary. For any two multi-indices μ μ μ and ννν of the same total degree, the scalar product of P_{μ μ μ}^{M} and P_{ν ν ν}^{M} is given by

(P_{μ μ μ}^{M}, P_{ν ν ν}^{M})_{d} = D^{ν ν ν}(P_{μ μ μ}^{M}) ,  (38)

where D^{ν ν ν} denotes ∏_{i<j} D_{ij}^{ν_{ij}}.

Proof. Let Π = Π_{a}^{(d)} be the projection from ℂ[S_{n}]_{a} onto the summand P_{a}(d) in the decomposition (9). (We will give an explicit formula for this projection operator in eq. (44) of §5 using the results of this section, but we do not need it here.) Then we have

(D_{ij}P, Q)_{d} = (P, t_{ij}Q)_{d} = (P, Π(t_{ij}Q))_{d} for P ∈ P_{a}(d) , Q ∈ P_{a−e_{i}−e_{j}}(d) ,  (39)

where the second equality follows from the definition of Π together with the second remark after Theorem 2, and the first from the first formula in (37) together with the fact that spherical polynomials are orthogonal to all polynomials of smaller degree (which is needed because the operator D_{ij}^{∗}, like E_{ij}^{∗}, is not homogeneous). On the other hand, for i ≠ j, the definition of the monomial basis implies that

P_{ν ν ν}^{M} = Π_{a}^{(d)}(t_{ij}P_{ν ν ν−e_{ij}}^{M})  for all ννν ∈ N_{0}(a) ,  (40)

since t_{ij}P_{ν ν ν−e_{ij}}^{M} differs from T^{ν ν ν} by something belonging to the ideal of ℂ[S_{n}]_{a} generated by the t_{kk} and since the decomposition (9) implies (by induction on the total degree) that ℂ[S_{n}]_{a} is the direct sum of P_{a}(d) and this ideal. Combining equations (40) and (39), we find

(P_{μ μ μ}^{M}, P_{ν ν ν}^{M})_{d} = (P_{μ μ μ}^{M}, Π(t_{ij}P_{ν ν ν−e_{ij}}^{M}))_{d} = (D_{ij}(P_{μ μ μ}^{M}), P_{ν ν ν−e_{ij}}^{M}))_{d} .

Applying this identity repeatedly we find finally that this scalar product equals (D^{ν ν ν}(P_{μ μ μ}), P_{0}^{M}). But D^{ν ν ν}(P_{μ μ μ}) is a constant (polynomial of degree 0), P_{0}^{M} = 1, and (1, 1)_{d} has been normalized to be equal to 1. □

Proposition 2 shows that each of the 3- or 4-dimensional subspaces ⟨D_{ij}, E_{ij}, E_{ji}, F_{ij}⟩ of which 𝔤 is composed is already closed under taking adjoints. More precisely, the elements

\begin{aligned}D_{ij}^{+} & = D_{ji}^{+} = D_{ij} − \frac{1}{2}(E_{ij} + E_{ji})  (1 ≤ i ≤ j ≤ n) , \\ F_{ij}^{+} & = F_{ji}^{+} = F_{ij}  (1 ≤ i ≤ j ≤ n) , \\ E_{ij}^{−} & = −E_{ji}^{−} = E_{ij} − E_{ji}  (1 ≤ i < j ≤ n) , \\ F_{ij}^{−} & = F_{ji}^{−} = F_{ij} − E_{ij} − E_{ji}  (1 ≤ i ≤ j ≤ n) \end{aligned}  (41)

give a basis of 𝔤 consisting of selfdual or anti-selfdual elements. In particular, the subspaces 𝔤^{±} = {X ∈ 𝔤 | X^{∗} = ±X} of 𝔤 have dimensions n^{2} + n and n^{2}, respectively, and 𝔤^{−} is a sub-Lie algebra of 𝔤, since [X, Y]^{∗} = −[X^{∗}, Y^{∗}]. More concretely we have

𝔤^{−} = \{( \begin{matrix}A & \frac{1}{2}(A + A^{t}) \\ 0 & −A^{t} \end{matrix} )\} ≅ 𝔤𝔩(n) .

Finally, we remark that instead of proving Propositions 1 and 2 for integral values of d ≥ n and deducing the general case by polynomiality, one could prove them directly for all d ∈ ℂ by using the description of ( , )_{d} in (24) together with the identity (again easily proved by integration by parts)

E_{d}^{𝔏}[s_{ij} F ] = E_{d}^{𝔏}[\frac{∂F}{∂s_{ij}} + δ_{ij} \frac{d − i}{s_{ii}} F ] for all F ∈ ℂ[𝔏_{n}]^{ev} .

However, this calculation is much more complicated than the one with the X-variables because the expressions for the operators E_{ij} and D_{ij} in terms of the S-variables are no longer polynomial, but involve powers of the diagonal elements s_{kk} of S in their denominators.

§5. The two canonical bases of P_{a}(d)

In §2 (Proposition 1 and the following remark) we observed that Theorem 1 implies the existence of a canonical basis {P_{ν ν ν}^{M}}_{ν ν ν∈N_{0}(a)} of P_{a}(d), characterized by the property that P_{ν ν ν}^{M}(T) becomes equal to T^{ν ν ν} if one sets all t_{ii} equal to 0. In this section we will study this monomial basis in detail. First, we will give an explicit construction of the polynomials P_{ν ν ν}^{M} in terms of a certain “harmonic projection operator” (Proposition 1). This at the same time gives a second, more constructive, proof of Theorem 1 and its corollary. Next, as an application of the Lie algebra 𝔤 of differential operators introduced above, we will give an inductive construction of the polynomials P_{ν ν ν}^{M} in terms of certain “raising operators” belonging to the universal enveloping algebra of 𝔤. We then discuss the dual basis {P_{ν ν ν}^{D}} of the basis {P_{ν ν ν}^{M}} with respect to the scalar product from §3 (for generic values of d ∈ ℂ) and show that its members satisfy (and in fact are determined by) the recursive property D_{ij}(P_{ν ν ν}^{D}) = P_{ν ν ν−e_{ij}}^{D}, where D_{ij} are the “mixed Laplace operators” from §4. Finally, again using the scalar product, we will show that each product t_{ij}P_{ν ν ν}^{M}(T) can be written as a linear combination with constant coefficients of a bounded number of basis elements P_{ν ν ν′}^{M}(T). This gives another recursive way to obtain the basis elements, generalizing the classical recursion relations for Legendre and Gegenbauer polynomials.

We start with the construction of the P_{ν ν ν}^{M} by a projection operator. Denote by H(ℝ^{d}) = Ker(Δ) ⊂ ℂ[ℝ^{d}] the subspace of harmonic polynomials on ℝ^{d}. As we mentioned before the corollary to Theorem 1 in §2, it is well-known that every polynomial P on ℝ^{d} can be written uniquely as

P (x) = ∑_{j≥0} P_{m}(x) (x, x)^{m}  (42)

where each P_{m} is harmonic (and homogeneous of degree a−2m if P is homogeneous of degree a). Denote by π^{(d)} : ℂ[ℝ^{d}] → H(ℝ^{d}) the projection map sending P to P_{0}, i.e., to the unique harmonic polynomial which is congruent to P modulo the ideal in ℂ[ℝ^{d}] generated by (x, x). The following formula for π^{(d)} is surely well-known, but we include its proof for lack of a convenient reference.

Lemma. Suppose that P ∈ ℂ[ℝ^{d}] is homogeneous of degree a. Then

π^{(d)}(P (x)) = ∑_{0≤j≤a/2} \frac{1}{4^{j} j! (2 − a − d/2)_{j}} Δ^{j}(P (x)) (x, x)^{j} ,  (43)

where (x)_{j} = x(x + 1) ⋯ (x + j − 1) is the ascending Pochhammer symbol.

Proof. We have Δ(P Q) = Δ(P)Q + 2 ∑_{α} \frac{∂P}{∂x_{α}} \frac{∂Q}{∂x_{α}} + P Δ(Q) for any P, Q ∈ ℂ[ℝ^{d}]. Apply this to Q(x) = (x, x)^{j}, for which we have

\frac{∂Q}{∂x_{α}} = 2j x_{α} (x, x)^{j−1} , \frac{∂^{2}Q}{∂x_{α}^{2}} = 2j (x, x)^{j−1} + 4j(j − 1) x_{α}^{2} (x, x)^{j−2} ,

to get

Δ(P (x)(x, x)^{j}) = Δ(P (x)) (x, x)^{j} + 4j (E(P (x)) + (d/2 + j − 1) P (x)) (x, x)^{j−1} ,

where E = ∑_{α} x_{α} \frac{∂}{∂x_{α}} is the Euler operator on ℂ[ℝ^{d}]. Replacing P by Δ^{j}(P) in this formula, where P is homogeneous of degree a, we find

Δ(Δ^{j}(P (x)) (x, x)^{j}) = Δ^{j+1}(P (x)) (x, x)^{j} + 4j (a + d/2 − j − 1) Δ^{j}(P (x)) (x, x)^{j−1} ,

since Δ^{j}(P) has degree a − 2j. It follows that the expression ∑ γ_{j} Δ^{j}(P (x)) (x, x)^{j} is in the kernel of Δ if γ_{j−1} + 4j(a + d/2 − j − 1)γ_{j} = 0 for all j ≥ 1, and combining this recursion with the condition γ_{0} = 1, which ensures that this expression is congruent to P (x) modulo (x, x), gives the formula stated. □

Now using the correspondence between the operators D_{i} on ℂ[S_{n}] and Δ_{i} on ℂ[(ℝ^{d})^{n}], we find from the lemma that for each a ∈ ℤ_{≥0}^{n} the product operator

Π = Π_{a}^{(d)} = ∏_{i=1}^{n}( ∑_{0≤j≤a_{i}/2} \frac{t_{ii}^{j} D_{i}^{j}}{4^{j} j! (2 − a_{i} − d/2)_{j}} )  (44)

(in which the order of the product does not matter since the operators for different i commute) sends any P ∈ ℂ[S_{n}]_{a} to the unique element of P_{a}(d) which is congruent to P modulo the ideal (t_{11}, …, t_{nn}) of ℂ[S_{n}], i.e., it gives the projection onto the μ μ μ = 0-component of the direct sum decomposition (9). (More precisely, the lemma gives this when d is an integer > n, and the general case follows by “analytic continuation” since it is equivalent to a collection of identities between rational functions of d, each true for infinitely many values of d.) In particular, we have the following extension of eq. (40):

Proposition 1. The monomial basis of P_{a}(d) for d ∈ ℂ ∖ Ξ(a) is given by

P_{ν ν ν,d}^{M} = Π_{a}^{(d)}(T^{ν ν ν})  (ν ∈ N_{0}(a)) ,  (45)

where Π_{a}^{(d)} : ℂ[S_{n}]_{a} → P_{a}(d) is the operator defined by equation (44).

We observe that the same argument as was used in the above calculation leads to a second proof of Theorem 1 of §2 and to an explicit form of the decomposition (44) of polynomials on S_{n}. Indeed, almost the same argument as the one used to prove (43) shows that the mth component P_{m}(x) of the decomposition (42) of a polynomial P ∈ ℂ[ℝ^{d}]_{a} is given by the formula

P_{m}(x) = \frac{1}{4^{m}m! (a + d/2 − 2m)_{m}} ∑_{0≤j≤a/2−m} \frac{(x, x)^{j} Δ^{m+j}(P (x))}{4^{j}j! (2m + 2 − a − d/2)_{j}} .  (46)

(One first checks that the expression on the right is annihilated by Δ either by repeating the calculation in the lemma or by observing that the sum is simply π^{(d)}(Δ^{m}P (x)), and then verifies the equality (42) for the polynomials defined by (46) using a simple binomial coefficient identity.^{4}) We immediately deduce:

Proposition 2. The projection from ℂ[S_{n}]_{a} to P_{a−2m} (0 ≤ m ≤ a/2) defined by the direct sum decomposition (9) is given explicitly by the operator

Π_{a,m}^{(d)} = ∏_{i=1}^{n}(\frac{1}{4^{m_{i}} m_{i}! (a_{i} + d/2 − 2m_{i})_{m_{i}}} ∑_{0≤j≤a_{i}/2−m_{i}} \frac{t_{ii}^{j} D_{i}^{m_{i}+j}}{4^{j} j! (2m_{i} + 2 − a_{i} − d/2)_{j}} )

and is, up to a scalar factor, equal to the map P ↦ Π_{a−2m}^{(d)}(D_{1}^{m_{1}} ⋯ D_{n}^{m_{n}}P ) with Π_{a}^{(d)} as in (40).

We now turn to the construction of the monomial basis, starting with the constant function P_{0}^{M} = 1, using “raising operators.” For all i ≠ j we define elements R_{ij} = R_{ji} in the universal enveloping algebra of 𝔤 by

R_{ij} = F_{ii}F_{jj}D_{ij} − 2(E_{jj} − 2)F_{ii}E_{ji} − 2(E_{ii} − 2)F_{jj}E_{ij} + 4(E_{ii} − 2)(E_{jj} − 2)F_{ij} . (47)

An alternative way to write this definition, since E_{ij}, E_{ji} and F_{ij} change the multidegree, is

R_{ij} = F_{ii}F_{jj}D_{ij} − 2F_{ii}E_{ji}(E_{jj} − 1) − 2F_{jj}E_{ij}(E_{ii} − 1) + 4F_{ij}(E_{ii} − 1)(E_{jj} − 1) . (48)

As operators on ℂ[S_{n}], these operators map ℂ[S_{n}]_{a} to ℂ[S_{n}]_{a+e_{i}+e_{j}} and in particular increase the total degree by 2. We will explain the origin of the formulas (47) and (48) below.

4namely (with n = m + j and A = a + d/2 − 1), that ∑_{m=0}^{n}(A − 2m)\binom{A}{m}\binom{2n−A}{n−m} = 0 if n > 0.

Theorem 5. The raising operators R_{ij} commute with one another and map the space P^{(n)}(d) of higher spherical polynomials to itself. For a ∈ (ℤ_{≥0})^{n} and ννν ∈ N_{0}(a) the monomial basis element P_{ν ν ν}^{M} ∈ P_{a}(d) is given by

P_{ν ν ν}^{M} = \frac{1}{ε_{2a}(d − 2)} R^{ν ν ν}(1)  (d ∈ ℂ, d ∉ 2 Z_{≤1}),  (49)

where ε_{2a}(d − 2) is defined by equations (14) and (18) and R^{ν ν ν} = ∏_{i<j} R_{ij}^{ν_{ij}} .

Proof. The identities

[R_{ij}, R_{kl}] = 0 ,  D_{k} R_{ij} = R_{ij}^{[k]} D_{k}  (i ≠ j, k ≠ l)  (50)

in the universal enveloping algebra of 𝔤, where R_{ij}^{[k]} is defined like R_{ij} but with E_{ii} − 2 and E_{jj} − 2 in (47) replaced by E_{ii} − 2 + 2δ_{ik} and E_{jj} − 2 + 2δ_{jk}, can be checked directly from the commutation relations given in §4. (However, this is very tedious. Two better ways will be given at the end of the proof.) These identities prove the first two statements of the theorem. For the last one, it suffices by induction to show that

R_{ij}(P_{ν ν ν}^{M}) = (d + 2a_{i} − 2) (d + 2a_{j} − 2) P_{ν ν ν+e_{ij}}^{M}  (51)

But this is almost obvious: The first three terms in (48) are divisible by t_{ii} or t_{jj} and the operator 4(E_{ii} − 1)(E_{jj} − 1) acts on ℂ[S_{n}]_{a} as multiplication by (d + 2a_{i} − 2)(d + 2a_{j} − 2), so

R_{ij}(P_{ν ν ν}^{M})(T)|_{t_{11}=···=t_{nn}=0} = (d + 2a_{i} − 2) (d + 2a_{j} − 2) t_{ij}P_{ν ν ν}^{M}(T)|_{t_{11}=···=t_{nn}=0}
= (d + 2a_{i} − 2) (d + 2a_{j} − 2) T^{ν ν ν+e_{ij}} ,

and now (51) follows from the definition of the monomial basis and the fact that R_{ij} sends P^{(n)}(d) to itself. □

The proof just given is short, but has two disadvantages: the definition (47) (or (48)) is completely unmotivated, and the commutation relations (50) are proved by a lengthy brute force calculation. We describe two other approaches that are more illuminating and also illustrate ideas that will be used again later.

(I) For i, j ∈ {1, …, n} and a ∈ ℂ^{n} we define an operator R_{ij}(a) by

R_{ij}(a) = δ(T)^{a+e_{i}+e_{j}−(2−d)1/2} D_{ij} δ(T)^{(2−d)1/2−a} ,  (52)

where δ(T)^{m} is defined as in the Corollary to Theorem 1. (Note that δ(T)^{m} is not well-defined for non-integral m, but the right-hand side of (52) is well-defined because the ambiguity of phase of the two δ(T)^{∗} terms cancel out.) This definition is motivated by the symmetry of the inhomogeneous form of the differential system defining P_{a}(d) under a ↦ (2 − d)1 − a, a symmetry which is implicit in Remark 4 of §3 and explicit in equation (8) in the case n = 3. A direct calculation shows that

P ∈ ℂ[S_{n}]_{a} ⇒ R_{ij}(a) P = \{ \begin{matrix}t_{ii}^{2}D_{i}(P) & if i = j, \\ R_{ij}(P) & if i ≠ j. \end{matrix}  (53)

This makes the fact that the operators R_{ij} commute almost obvious, because applying R_{ij} to ℂ[S_{n}]_{a} is the same as applying D_{ij} to δ(T)^{(2−d)1/2−a}ℂ[S_{n}]_{a}, and the operators D_{ij} all commute with one another, and an exactly similar argument shows that the operators R_{ij} map spherical polynomials to spherical polynomials, because (53) shows that the spherical polynomials in ℂ[S_{n}]_{a} are precisely the polynomials killed by all R_{kk}(a).

(II) The second approach is to prove equation (51) directly using the harmonic projection operators Π_{a}^{(d)} discussed in the first part of this section. This equation then implies equation (49) and the facts that the operators R_{ij} preserve P^{(n)}(d) and (at least when restricted to this space) that they commute with one another (because ννν + e_{ij} + e_{kl} = ννν + e_{kl} + e_{ij}). Because of equation (40), all we have to do to establish (51) (in which we take i = 1, j = 2 for convenience) is to show that

\begin{matrix}(d + 2a_{1} − 2)(d + 2a_{2} − 2) Π_{a}^{(d)}(t_{12}P ) = 4(d + 2a_{1} − 2)(d + 2a_{2} − 2)t_{12}P \\ − 2(d + 2a_{2} − 2)t_{11}E_{21}(P) − 2(d + 2a_{1} − 2)t_{22}E_{12}(P) + t_{11}t_{22} D_{12}(P) \end{matrix}  (54)

for all P ∈ P_{a}(d). We do this using the commutation relations given in Theorem 4.

Denote by π_{k} = π_{k}^{(d)} the kth factor in (44), so that Π_{a}^{(d)} = π_{1} ⋯ π_{n}. Since π_{k} commutes with F_{12} for k ≥ 3 and projects P to itself, we have Π_{a}^{(d)}(t_{12}P )= π_{1}π_{2}(t_{12}P ). From the commutation relations given in Theorem 4 and the fact that D_{i} = D_{ii} annihilates P for all i, we find

D_{2}(t_{12}P ) = [D_{2}, F_{12}](P) = 2 E_{12}(P) , D_{2}^{2}(t_{12}P ) = 2 [D_{2}, E_{12}](P) = 0 ,

so that only the first two terms in the series defining π_{2} contribute to π_{2}(t_{12}P) and we have

π_{2}(t_{12}P ) = t_{12}P − \frac{1}{2(2a_{2} + d − 2)} t_{22}E_{12}(P) .

By the same argument, and since π_{1} commutes with F_{22}, we find

π_{1}π_{2}(t_{12}P ) = F_{12}(P) − \frac{1}{2(2a_{1} + d − 2)} t_{11}E_{21}(P) − \frac{1}{2(2a_{2} + d − 2)} t_{22} π_{1}(E_{12}P ) ,

and since an exactly similar argument using the commutation relation [D_{1}, E_{12}] = 2D_{12} gives

π_{1}(E_{12}P ) = E_{12}(P) − \frac{1}{2(a_{1} + d − 2)} t_{11}D_{12}(P) ,

this completes the proof of (54) and hence also of (49). It also explains the motivation for the artificial-looking definition of R_{ij} : this operator is the simplest element of the universal enveloping algebra of the Lie algebra of §4 that is (up to a scalar factor in each homogeneous component) adjoint to the mixed Laplace operator D_{ij} with respect to the restriction of the scalar product ( · , ·)_{d} to P(d).

We now come to the definition of the second canonical basis of P^{(n)}(d). Theorem 5 tells us that, for generic d, the monomial basis {P_{ν ν ν}^{M}(T) | ννν ∈ N_{0}(a)} of P^{(n)}(d) is the same, up to scalar factors, as the “ascending basis” defined by taking 1 as the basis element of P_{0}(d) and then applying the raising operators R_{ij} to obtain the higher basis elements by induction. The descending basis is defined, again for generic d, by a similar process in reverse.

Proposition-Definition. For generic d, there exists a unique basis {P_{ν ν ν}^{D}(T) | ννν ∈ N_{0}(a)} of P^{(n)}(d), called the descending basis, characterized by the property

D_{ij}(P_{ν ν ν}^{D}) = P_{ν ν ν−e_{ij}}^{D}  (i ≠ j)  (55)

(where the right-hand side is to be taken as 0 if ν_{ij} = 0) and the initial condition P_{0}^{D}(T) = 1.

This follows immediately from the following lemma, in which P_{a}(d) = {0} if a_{i} < 0 for some i.

Lemma. Let a ≠ 0 and d a complex number such that the inner product in P_{a}(d) is nondegenerate. Suppose that for all i ≠ j we are given polynomials G_{ij} = G_{ji} ∈ P_{a−e_{i}−e_{j}}(d) such that D_{kl}(G_{ij}) = D_{ij}(G_{kl}) for all i ≠ j and k ≠ l. Then there exists a unique polynomial G ∈ P_{a}(d) such that D_{ij}(G) = G_{ij} for all i ≠ j.

Proof. The proof is similar to that of Theorem 1. We will show by induction on |S| that, for any set S of (unordered) pairs (i, j) with 1 ≤ i, j ≤ n, i ≠ j, the following statements hold:

(i) The dimension of the space K_{a}(S) := ⋂_{(i,j)∈S} Ker(D_{ij}), where Ker(D_{ij}) = {P ∈ P_{a}(d) | D_{ij}P = 0}, is equal to the number of ννν ∈ N_{0}(a) with ν_{ij} = 0 for all (i, j) ∈ S.

(ii) The map D_{ij} : K_{a}(S) → K_{a−e_{i}−e_{j}}(S) is surjective whenever (i, j) ∉ S.

(iii) Assume that for all (i, j) ∈ S, polynomials G_{ij} ∈ P_{a−e_{i}−e_{j}}(d) are given which satisfy D_{kl}(G_{ij}) = D_{ij}(G_{kl}) for all (k, l) ∈ S. Then there exists G ∈ P_{a}(d) such that D_{ij}(G) = G_{ij} for all (i, j) ∈ S.

We denote by Π = Π_{a}^{(d)} the harmonic projection. We saw earlier (eq. (39)) that the scalar product (Q, D_{ij}P)_{d} equals (Π(t_{ij}Q), P)_{d} for any P in P_{a}(d) and Q in P_{a−e_{i}−e_{j}}(d). It follows that D_{ij}P = 0 if and only if P is orthogonal to all monomial basis polynomials P_{ν ν ν}^{M} with ν_{ij} > 0, and hence that the dimension of K_{a}(S) equals N_{0}(a) minus the number of ννν ∈ N_{0}(a) with ν_{ij} > 0 for some pair (i, j) ∈ S, which is precisely the statement of (i). Statement (ii) follows immediately from (i) by comparing the dimensions. We now prove (iii) by induction on the cardinality of S. The claim is true when |S| = 1 since D_{ij} : P_{a}(d) → P_{a−e_{i}−e_{j}}(d) is surjective by (ii). Now assume that (iii) is true for some S and that G_{ij} are given for (i, j) ∈ S∪{(k, l)} and satisfy the compatibility condition. By the inductive assumption, there exists G_{0} ∈ P_{a}(d) such that D_{ij}(G_{0}) = G_{ij} for any (i, j) ∈ S. We have D_{ij}(G_{kl} − D_{kl}(G_{0})) = D_{kl}(G_{ij} − D_{ij}(G_{0})) = 0, so G_{kl} − D_{kl}(G_{0}) ∈ K_{a−e_{k}−e_{l}}(S). Since D_{kl} maps K_{a}(S) to K_{a−e_{k}−e_{l}}(S) surjectively by (ii), there exists G_{1} ∈ K_{a}(S) such that D_{kl}(G_{1}) = G_{kl} − D_{kl}(G_{0}). So if we set G = G_{0} + G_{1} then D_{kl}(G) = G_{kl} and, since D_{ij}G_{1} = 0 for any (i, j) ∈ S, also D_{ij}(G) = G_{ij}. The uniqueness of G follows from (i), which implies that the dimension of K_{a}(S) is 0 if S is the set of all (i, j) with i ≠ j and a ≠ 0. □

Remark. The Lemma, and therefore also the Proposition-Definition, apply only to generic values of d for which the scalar product on P_{a}(d) is non-degenerate. In fact this assumption holds for all d ∉ ℤ_{<n}. This will follow from the results in §9, where we will give an independent proof of the existence of {P_{ν ν ν}^{D}} satisfying (55) using a generating function. To prove it directly, we would need an a priori proof that ⋂_{i,j} Ker(D_{ij}) = {0} for d ∉ ℤ. We were not able to give this, since the argument used for Theorem 1 does not generalize in any obvious way. For example, for a = (1, 1, 1, 1), then the space of harmonic polynomials is spanned by the three monomials t_{12}t_{34}, t_{13}t_{24} and t_{14}t_{23}; their images under D_{12} are t_{34} times d, 1 and 1, respectively, so that there would seem to be a problem only if d = 0, but in fact the trouble occurs for the two values d = −2 (where there is a 1-dimensional radical, with basis the sum of the three polynomials above) and d = 1 (where the radical is 2-dimensional, spanned by their differences).

Theorem 6. The monomial and descending bases of P_{a}(d) are dual to one another with respect to the scalar product.

Proof. Let μ μ μ, ννν ∈ N_{0}. We have to show that (P_{ν ν ν}^{M}, P_{μ μ μ}^{D})_{d} = δ_{ν ν ν,μ μ μ}. We can assume that ννν and μ μ μ have the same multidegree a, since otherwise the inner product vanishes automatically. We proceed by induction on the total degree. If a = 0, then P_{ν ν ν}^{M} = P_{μ μ μ}^{D} = 1 and (1, 1)_{d} = 1 = δ_{ν ν ν,μ μ μ}. Otherwise, we have ν_{i,j} > 0 for some i ≠ j and we find, using the properties of the projection operator Π, equation (39) and the induction assumption,

(P_{ν ν ν}^{M}, P_{μ μ μ}^{D})_{d} = (Π(t_{ij}P_{ν ν ν−e_{ij}}^{M}), P_{μ μ μ}^{D})_{d} = (t_{ij}P_{ν ν ν−e_{ij}}^{M}, P_{μ μ μ}^{D})_{d}
= (P_{ν ν ν−e_{ij}}^{M}, D_{ij}P_{μ μ μ}^{D})_{d} = (P_{ν ν ν−e_{ij}}^{M}, P_{μ μ μ−e_{ij}}^{D})_{d} = δ_{ν ν ν,μ μ μ} . □

In the preceding theorem we did not specify what d was. In fact there are two possible interpretations. One is to assume, as we did in the preceding Proposition-Definition and Lemma, that d is a specific complex number for which the scalar product is non-degenerate (i.e., in view of the Remark above, that d ∉ ℤ_{<n}). The other is to consider d as a variable. In that case the coefficients of the canonical basis elements P_{ν ν ν}^{M} and P_{ν ν ν}^{D}, and the coefficients of the scalar products (X, Y)_{d} for any X, Y ∈ ℚ[S_{n}], are elements of the field K := ℚ(d), and Theorem 6 becomes an identity over the field K. To make this clearer, take the example n = 4, a = (1, 1, 1, 1), considered above. Here N_{0}(a) = 3, N_{0}(a) consists of the three 4 × 4 matrices ννν_{1} = e_{12} + e_{34}, ννν_{2} = e_{13} + e_{24}, ννν_{3} = e_{14} + e_{23}, and the space P_{a}(d) is spanned by the three monomials P_{1}(T) = t_{12}t_{34}, P_{2}(T) = t_{13}t_{24} and P_{3}(T) = t_{14}t_{23}. (This is obviously the monomial basis, which here consists of actual monomials.) The corresponding Gram matrix is given by

(P_{i}, P_{j})_{1≤i, j≤3} = ( \begin{matrix}d^{2} & d & d \\ d & d^{2} & d \\ d & d & d^{2} \end{matrix} )  (56)

and the descending basis by

( \begin{matrix}P_{1}^{D}(T) \\ P_{2}^{D}(T) \\ P_{3}^{D}(T) \end{matrix} ) = = \frac{1}{d(d − 1)(d + 2)} ( \begin{matrix}d + 1 & −1 & −1 \\ −1 & d + 1 & −1 \\ −1 & −1 & d + 1 \end{matrix} ) ( \begin{matrix}P_{1}(T) \\ P_{2}(T) \\ P_{3}(T) \end{matrix} ) ,

in which, by virtue of Theorem 6, the 3 × 3 matrix is just the inverse of the matrix in (56). Many more examples can be found in Table 2 in the appendix at the end of the paper.

We make one final observation. Define degree-preserving operators C_{ij} (i, j = 1, …, n) by

C_{ij} = F_{ij}D_{ij} − E_{ij}E_{ji} + E_{ii} .

From the commutation relations given in Theorem 2, we find that C_{ij} = C_{ji} (because [E_{ij}, E_{ji}] = E_{ii} −E_{jj}) and that the C_{ij} commute with all D_{k}. The latter property implies that the operators C_{ij} act on each space P_{a}(d). For i = j the operator C_{ii} = F_{ii}D_{i} − E_{ii}(E_{ii} − 1) corresponds to the usual Casimir operator in the Lie algebra 𝔰𝔩(2) generated by the three operators D_{i}, E_{ii} and F_{ii} on (ℝ^{d})_{i}, so we can think of the C_{ij} as a kind of “mixed Casimir operators.” On P_{a}(d), the operator C_{ii} is equal to the scalar −(a_{i} + d/2)(a_{i} + d/2 − 1) and the mixed operator C_{ij} is given by the commutator relation

[D_{ij}, R_{ij}] = 4(a_{i} + a_{j} − 2) (C_{ij} − a_{i} − a_{j} + a_{i}a_{j}) ,

but the operators C_{ij} for i ≠ j do not act as scalars and do not even commute with each other. We do not know whether these operators have interesting applications.

§6. Higher spherical polynomials and invariant harmonic polynomials

In this section we look in more detail at what happens with the spaces P^{(n)}(d) when d is a positive integer smaller than n.

Our original motivation for the definition of P^{(n)}(d) was to consider functions P (T) whose pull-back P̃ = P ∘ β_{n} to (ℝ^{d})^{n} is harmonic in each component x_{i} ∈ ℝ^{d}. Because the map β_{n}^{∗} : P ↦ P̃ is not injective for d < n, we did not use this as the definition of P^{(n)}(d) in that case, but instead defined P^{(n)}(d) as ⋂_{i=1}^{n} Ker(D_{i}) . In this section we study the relationship of the three vector spaces

(i) P_{a}(d) = {P ∈ ℂ[S_{n}]_{a} | D_{1}P = ⋯ = D_{n}P = 0} ,
(ii) H_{a}(d) = (H_{a_{1}}(ℝ^{d}) ⊗ … ⊗ H_{a_{n}}(ℝ^{d}))^{O(d)} ,
(iii) V_{a}(d) = {P ∈ ℂ[S_{n}]_{a} | P̃ ∈ H_{a}(d)} ,

where H_{a_{i}}(ℝ^{d}) in (ii) is defined as in Remark 3 of §2. For d ≥ n, all three spaces are isomorphic. In general, we have P_{a}(d) ⊆ V_{a}(d) (because Δ_{i}(P̃) = D_{i}(P)^{∼}), and we will show that the map β_{n}^{∗} : P_{a}(d) → H_{a}(d) is surjective (Theorem 7). In particular, we have

dim H_{a}(d) ≤ dim P_{a}(d) ≤ dim V_{a}(d) .  (57)

At the end of this section we will show that in general all three dimensions are distinct, and will determine all three of them exactly in the special case when d = n − 1.

Let us define two further spaces

K_{a}(d) = Ker(β_{n}^{∗} : ℂ[S_{n}]_{a} → (ℂ[ℝ^{d}]_{a_{1}} ⊗ ⋯ ⊗ ℂ[ℝ^{d}]_{a_{n}})^{O(d)} ) ,
K′_{a}(d) = K_{a}(d) ∩ P_{a}(d) .

Clearly K_{a}(d) ⊆ V_{a}(d).

Theorem 7. The diagram

0 → K_{a}(d) ⟶ ℂ[S_{n}]_{a} ⟶^{β_{n}^{∗}} (⨂_{i=1}^{n} ℂ[ℝ^{d}]_{a_{i}})^{O(d)} ⟶ 0
∥  ⋃  ⋃
0 → K_{a}(d) ⟶ V_{a}(d) ⟶^{β_{n}^{∗}}  H_{a}(d)  ⟶ 0
⋃  ⋃  ∥
0 → K′_{a}(d) ⟶ P_{a}(d) ⟶^{β_{n}^{∗}}  H_{a}(d)  ⟶ 0

is commutative with exact rows.

Proof. Only the surjectivity of β_{n}^{∗} in each of the three rows has to be proved, since the spaces on the left are by definition the kernel of β_{n}^{∗} in each case. The surjectivity of β_{n}^{∗} in the first row is a classical result (“first fundamental theorem of invariant theory,” [16]), and the surjectivity in the second row an immediate consequence of this, since V_{a}(d) is defined as the inverse image of H_{a}(d) in ℂ[S_{n}]_{a} . Thus the real assertion of the theorem is the statement that the map β_{n}^{∗} : P_{a}(d) → H_{a}(d) is surjective for all n and d (or equivalently, that the space V_{a}(d) is the sum of its two subspaces K_{a}(d) and P_{a}(d)). We prove this using the two following lemmas.

Lemma 1. For any P ∈ ℂ[S_{n}]_{a} and m ∈ ℤ_{≥0}^{n} we have

D_{i}(δ(T)^{m} P (T)) = δ(T)^{m} D_{i}(P (T)) + 2m_{i}(d+2a_{i}+2m_{i}−2) δ(T)^{m−e_{i}} P (T) (i = 1, …, n) .

Proof. This follows by induction on the non-negative integer m_{i} from the commutation relations [D_{i}, F_{ii}] = 4E_{ii} = 4E_{i} + 2d and [D_{i}, F_{jj}] = 0 (i ≠ j). □

Lemma 2. Let P = ∑_{m} δ(T)^{m} P_{m}(T) be the decomposition of a polynomial P ∈ ℂ[S_{n}] given by the corollary to Theorem 1, where d is a positive integer. If P̃ = 0, then P̃_{m} = 0 for every m.

Proof. This follows immediately from the commutativity of the diagram

ℂ[S_{n}]_{a}  ⟵^{≅}  ⨁_{0≤m≤ \frac{1}{2} a} P_{a−2m}(d)
β_{n}^{∗} ↓  ↓ β_{n}^{∗}
ℂ[ℝ^{d}]_{a_{1}} ⊗ ⋯ ⊗ ℂ[ℝ^{d}]_{a_{n}}  ⟵^{≅} ⨁_{0≤m≤ \frac{1}{2} a} H_{a_{1}−2m_{1}}(ℝ^{d}) ⊗ ⋯ ⊗ H_{a_{n}−2m_{n}}(ℝ^{d})

31 in which the horizontal isomorphisms are those given by the decompositions (9) and (42). □

Returning to the proof of Theorem 7, let F ∈ H_{a}(d) and choose P ∈ ℂ[S_{n}]_{a} with P̃ = F. By definition we have P ∈ V_{a}(d) and by the relationship between D_{i} and the Laplacian Δ_{i} on (ℝ^{d})_{i} discussed in §1 we have (D_{i}P)^{∼} = Δ_{i}(P̃) = Δ_{i}(F) = 0 for all i. Let P = ∑_{m} δ(T)^{m} P_{m}(T) be the decomposition of P given by (9). Applying D_{i} to both sides of this equality and using Lemma 1 (with P replaced by P_{m} and a by a − 2m), we find

D_{i}(P) = ∑_{\begin{matrix}m≥0 \\ m_{i}≥1 \end{matrix}} 2m_{i} (d + 2a_{i} − 2m_{i} − 2) δ(T)^{m−e_{i}} P_{m}(T) .

The fact that D_{i}(P)^{∼} = 0 and Lemma 2 then imply that P̃_{m} = 0 for all m with m_{i} > 0, because the factor 2m_{i}(d + 2a_{i} − 2m_{i} − 2) is non-zero for d > 0 and 0 < m_{i} ≤ a_{i}/2, and since this is true for every i, it follows that P̃_{m} = 0 for all m ≠ 0, so F = P̃ = P̃_{0} ∈ β_{n}^{∗}(P_{a}(d)). □

From Theorem 7 we see that V_{a}(d) = K_{a}(d) + P_{a}(d) and hence that

dim H_{a}(d) = dim V_{a}(d) − dim K_{a}(d) = dim P_{a}(d) − dim K′_{a}(d)  (58)

so that the claim that both inequalities in (57) are in general strict is equivalent to the claim that both inclusions 0 ⊆ K′_{a}(d) ⊆ K_{a}(d) are in general proper. We will prove this by computing all three spaces in question in the special case d = n − 1.

Proposition 1. For d = n − 1 we have the dimension formulas

dim H_{a}(n − 1) = N_{0}(a) − N_{0}(a − 2) ,
dim P_{a}(n − 1) = N_{0}(a) ,
dim V_{a}(n − 1) = N_{0}(a) + N(a − 2) − N_{0}(a − 2) ,

where 2 = 2 · 1 = (2, …, 2) .

Proof. In general, it is known (“second fundamental theorem of invariant theory”) that for d < n the kernel of the map β_{n}^{∗} is the ideal in ℂ[S_{n}] generated by all (d + 1) × (d + 1) minors of the coordinate T ∈ S_{n}. In particular, if d = n − 1 then this kernel is the ideal generated by the single polynomial D ∈ ℂ[S_{n}]_{2} defined by D(T) = det T. In other words, we have

K_{a}(n − 1) = D · ℂ[S_{n}]_{a−2} .  (59)

To complete the proof, we will show that

K′_{a}(n − 1) = D · P_{a−2}(n + 3) .  (60)

Equations (59) and (60) and Theorem 1 give dim K_{a}(n − 1) = N(a − 2) and dim K′_{a}(n − 1) = N_{0}(a − 2), and together with the exact sequences of Theorem 7 these imply the dimension formulas given in the proposition.

To prove (60), we have to compute the effect of the ith Laplace operator D_{i} on products of the form P = D · Q. Since the formulas depend on the value of d, we will temporarily write D_{i}^{(d)} rather than simply D_{i} to denote the differential operator on ℂ[S_{n}] defined by (5).

Lemma 3. We have

D_{i}^{(n−1)}(D) = 0  (1 ≤ i ≤ n)  (61)

and more generally

D_{i}^{(n−1)}(D Q) = D D_{i}^{(n+3)}(Q)  (1 ≤ i ≤ n)  (62)

for any polynomial Q ∈ ℂ[S_{n}] .

Proof. The partial derivatives of D are easily seen to be given by

∂_{ij}D = 2 D_{ij} (1 ≤ i, j ≤ n)

where D_{ij}(T) is the (i, j)-cofactor of T. (In checking this for i = j, one has to remember that ∂_{ii} = 2∂/∂t_{ii}.) Using this and the formula

∑_{k=1}^{n} t_{jk} D_{ik} = δ_{ij} D

(the product of T and its adjoint matrix equals det(T) times the identity matrix), we find

D_{i}^{(d)}(D) = 2d D_{ii} + ∑_{j, k=1}^{n} t_{jk} ∂_{ij}( 2 D_{ik})

= 2d D_{ii} + 2 ∑_{j, k=1}^{n}(∂_{ij}(t_{jk} D_{ik}) − ∂_{ij}(t_{jk}) D_{ik})

= 2d D_{ii} + 2 ∑_{j=1}^{n} ∂_{ij}(∑_{k=1}^{n} t_{jk} D_{ik}) − 2 ∑_{j, k=1}^{n} δ_{ik}(1 + δ_{ij}) D_{ik}

= 2d D_{ii} + 4 D_{ii} − 2 ∑_{j=1}^{n} (1 + δ_{ij}) D_{ii}

= 2(d + 1 − n) D_{ii} ,

and assertion (61) follows. (Another proof, more in keeping with the contents of this section, is as follows. Let D = ∑_{0≤m≤1} δ(T)^{m} D_{m} with D_{m} ∈ P_{2−2m}(d−1) be the canonical decomposition of D as given by Theorem 1. From D̃ = 0 and Lemma 2 we deduce that D̃_{m} = 0 for all m. But then D_{m} ∈ Ker(β_{n}^{∗}) = D · ℂ[S_{n}], and since the degree of D_{m} is smaller than that of D for m ≠ 0, this implies that D_{m} = 0 for all m ≠ 0 and hence that D = D_{0} ∈ P_{2}(n − 1) .) To prove the second assertion, we apply the easily proved general formula

D_{i}^{(d)}(P Q) = D_{i}^{(d)}(P) Q + 2 ∑_{j,k=1}^{n} t_{jk} ∂_{ij}(P) ∂_{ik}(Q) + P D_{i}^{(d)}(Q)

with d = n − 1, P = D and use (61) to find

D_{i}^{(n−1)}(D Q) = 2 ∑_{k=1}^{n}(2 ∑_{j=1}^{n} t_{jk} D_{ij}) ∂_{ik}(Q) + D D_{i}^{(n−1)}(Q)
= D (D_{i}^{(n−1)}(Q) + 4∂_{ii}(Q)) = D D_{i}^{(n+3)}(Q) . □

Formula (60) follows immediately: If P ∈ K_{a}(n − 1), we can write P = D Q with Q ∈ ℂ[S_{n}]_{a−2}, and equation (62) shows that P ∈ P_{a}(n − 1) if and only if Q ∈ P_{a−2}(n + 3). □

We end this section with a final remark. As we mentioned before the Corollary of Theorem 1 and have used several times, given a non-degenerate quadratic form Q(x) = (x, x) on ℝ^{d} we have a direct sum decomposition ℂ[ℝ^{d}] = ⨁_{m≥0} Q(x)^{m} H(ℝ^{d}) of the space ℂ[ℝ^{d}] and hence a direct sum decomposition

ℂ[ℝ^{d}]^{⊗n} = ⨁_{m≥0} (∏_{i=1}^{n} Q(x_{i})^{m_{i}}) H(ℝ^{d})^{⊗n} = H(ℝ^{d})^{⊗n} ⊕ I_{n} ,

of its nth tensor power, where I_{n} is the ideal generated by the polynomials Q(x_{1}), …, Q(x_{n}) in ℂ[ℝ^{nd}] = ℂ[x_{1}, …, x_{n}], the splitting being given explicitly by the product of the projection operators π_{i}^{(d)} (1 ≤ i ≤ n) defined in (43). In particular, a polynomial P (x_{1}, …, x_{n}) that is harmonic with respect to each variable x_{i} ∈ ℝ^{d} and that belongs to the ideal I_{n} vanishes. But, at least for d ≥ 3, the elements of I_{n} are precisely the polynomials vanishing on the discriminant variety D_{n} = {(x_{1}, …, x_{n}) ∈ ℂ^{dn} | Q(x_{1}) = ⋯ = Q(x_{n}) = 0}. (Proof: The quadratic form Q is irreducible because a reducible form Q(x) = (a^{t}x)(b^{t}x) corresponds to the symmetric matrix a^{t}b + ab^{t} of rank ≤ 2, so cannot be non-degenerate if d ≥ 3. But then D_{n} is the product of n irreducible varieties and hence irreducible, so its associated ideal I_{n} is prime and hence equal to its own radical. The assertion then follows from Hilbert’s Nullstellensatz.) This proves the following statement, which will be used at the end of the next section in the case when n ≥ d = 4.

Proposition 2. A polynomial belonging to H(ℝ^{d})^{⊗n} is completely determined by its restriction to the set {Q(x_{1}) = ⋯ = Q(x_{n}) = 0} ⊂ ℂ^{dn}. In particular, if this restriction is O(d)-invariant, then so is the original polynomial.

Chapter II. Explicit Constructions

§7. Construction of invariant harmonic polynomials for d = 4

In this section we will give a construction of higher spherical polynomials for the special case d = 4 by identifying ℂ^{d} in this case with the space M(2, ℂ) of 2 × 2 complex matrices,^{5} with the quadratic form g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ↦ det(g) = ad − bc. The group G = SL(2, ℂ) acts on M(2, ℂ) by both right and left multiplication, and the combined action of G × G on M(2, ℂ) ≅ ℂ^{4} identifies (G × G)/{±1} with the special orthogonal group SO(4).

Let V_{1} denote the standard 2-dimensional representation of G and V_{a} (a ∈ ℕ) its ath symmetric power, the (a + 1)-dimensional space of homogeneous polynomials of degree a in two variables x and y. We have a G-invariant scalar product in V_{a} given by

⟨x^{p}y^{q}, x^{p′}y^{q′}⟩ = (−1)^{p} p! q! δ_{p,q′}  (p + q = p′ + q′ = a) ,  (63)

so we obtain by tensor product a G^{n}-invariant scalar product ⟨ , ⟩ on V_{a} = V_{a_{1}} ⊗ ⋯ ⊗ V_{a_{n}}. Now for each multi-index ννν ∈ N_{0}(a) we have the G-invariant vector

w_{ν ν ν} = ∏_{1≤i<j≤n} \frac{(x_{i}y_{j} − x_{j}y_{i})^{ν_{ij}}}{ν_{ij}!} ∈ V_{a} ,

where (x_{i}, y_{i}) are the coordinates on V_{a_{i}}. We define a function F_{ν ν ν} on M_{2}(ℂ)^{n} by

F_{ν ν ν}(g) = ⟨gw_{ν ν ν}, w_{ν ν ν}⟩ ( g = (g_{1}, …, g_{n}) ∈ M_{2}(ℂ)^{n} ) ,

and more generally set F_{μ μ μ,ν ν ν}(g) = ⟨gw_{μ μ μ}, w_{ν ν ν}⟩ for any μ μ μ, ννν ∈ N_{0}(a).

Proposition 1. The polynomial F_{μ μ μ,ν ν ν} is a homogeneous harmonic polynomial of degree a_{i} with respect to g_{i} ∈ M_{2}(ℂ) ≅ ℂ^{4} for each index i = 1, …, n. It is SO(4)-invariant for any μ μ μ and ννν, and is O(4)-invariant if μ μ μ = ννν.

Proof. The homogeneity property is obvious. For the harmonicity we must show that Δ_{i}F_{μ μ μ,ν ν ν} = 0 for each i, where g_{i} = ( \begin{matrix}a_{i} & b_{i} \\ c_{i} & d_{i} \end{matrix} ), Δ_{i} = 4(\frac{∂^{2}}{∂a_{i}∂d_{i}} − \frac{∂^{2}}{∂b_{i}∂c_{i}} ). This follows from the observation that Δ_{i} annihilates F (a_{i}x + b_{i}y, c_{i}x + d_{i}y) for any twice differentiable function F (x, y), whence Δ_{i}(gw_{μ μ μ}) = 0 and a fortiori Δ_{i}(⟨gw_{μ μ μ}, w_{ν ν ν}⟩) = 0 for any ννν. Finally, since w_{ν ν ν} is (left) invariant under the diagonal action of G on V_{a} and the scalar product ⟨ , ⟩ is G-invariant, it is clear that F_{μ μ μ,ν ν ν}(g) = ⟨gw_{μ μ μ}, w_{ν ν ν}⟩ is invariant under both right and left multiplication of g by elements of G, i.e., it is invariant under the action of SO(4). The G-invariance and (−1)^{a}-symmetry of the scalar product on V_{a} and the assumption that a_{1}+⋯+a_{n} is even imply that F_{μ μ μ,ν ν ν}(g^{∗}) = F_{ν ν ν,μ μ μ}(g), where g ↦ g^{∗} is the involution on M_{2}(ℂ)^{n} induced by the involution ∗ : ( \begin{matrix}a & b \\ c & d \end{matrix} ) ↦ ( \begin{matrix}d & −b \\ −c & a \end{matrix} ) of M_{2}(ℂ). Since this involution represents the non-trivial coset of SO(4) in O(4), it follows that the polynomials F_{μ μ μ,ν ν ν} + F_{ν ν ν,μ μ μ}, and hence in particular also the polynomials F_{ν ν ν} = F_{ν ν ν,ν ν ν}, are O(4)- invariant. □

We next define a generating function which has all of the polynomials F_{μ μ μ,ν ν ν}(g) as its coefficients. Let U = (U_{ij}) and V = (V_{ij}) be antisymmetric n × n matrices with variable coefficients (so that each is coordinatized by n(n − 1)/2 independent variables U_{ij}, V_{ij} with i < j) and set

F (g; U, V) = ∑_{μ μ μ, ν ν ν} F_{μ μ μ,ν ν ν}(g) U^{μ μ μ}V^{ν ν ν} ,

5Here we work over ℂ rather than ℝ (cf. footnote 2 in §1) because the determinant form on 2 × 2 matrices is not isomorphic to the standard quadratic form ∑_{1}^{4} x_{i}^{2} over ℝ, but is over ℂ.

where U^{μ μ μ} means ∏_{i<j} U_{ij}^{μ_{ij}} and we set F_{μ μ μ,ν ν ν} equal to 0 if μ μ μ and ννν have different multidegrees. The homogeneity of the F_{μ μ μ,ν ν ν} implies that this generating function is unchanged if we replace g_{i} by g_{i}/λ_{i} and U_{ij} by λ_{i}λ_{j}U_{ij} for any non-zero constants λ_{i}, so we may restrict ourselves to the case when all the g_{i} are unimodular.

Theorem 8. The generating function F (g; U, V) for g ∈ SL(2, ℝ)^{n} is given by

F (g; U, V) = D(g; U, V)^{−1/2} ,

where D(g; U, V), a polynomial in the coefficients a_{i}, b_{i}, c_{i}, d_{i} of g and U_{ij}, V_{ij} of U and V, is defined as the determinant of the symmetric 4n × 4n matrix

( \begin{matrix}0_{n} & U & a & b \\ −U & 0_{n} & c & d \\ a & c & 0_{n} & V \\ b & d & −V & 0_{n} \end{matrix} ) ,

where a, b, c, d are the n × n diagonal matrices with diagonal entries a_{i}, b_{i}, c_{i}, d_{i}, respectively.

Proof. Extend the scalar product (63) to be 0 for polynomials of different degree, i.e., replace the right-hand side of (63) by ⟨x^{p}y^{q}, x^{p′}y^{q′}⟩ = (−1)^{p}p!q!δ_{pq′}δ_{p′q}. Then ⟨u, v⟩ = 0 for u ∈ W_{ν ν ν}, v ∈ W_{μ μ μ} with ννν·1 ≠ μ μ μ·1, so that the definition of F_{μ μ μ,ν ν ν}(g) as ⟨gw_{μ μ μ}, w_{ν ν ν}⟩ holds for all μ μ μ and ννν. From the definition of w_{μ μ μ} we have

∑_{μ μ μ} w_{μ μ μ} U^{μ μ μ} = ∏_{1≤i<j≤n} e^{U_{ij}(x_{i}y_{j}−x_{j}y_{i})} = exp(∑_{i,j} U_{ij}x_{i}y_{j}) ,

and hence

F (g; U, V) = ∑_{μ μ μ,ν ν ν}⟨w_{μ μ μ}^{∗}, w_{ν ν ν}⟩ U^{μ μ μ}V^{ν ν ν} = ⟨exp(∑_{i,j} U_{ij}x_{i}^{∗}y_{j}^{∗}), exp(∑_{i,j} V_{ij}x_{i}y_{j})⟩ ,

where ( \begin{matrix}x_{i}^{∗} \\ y_{i}^{∗} \end{matrix} ) := g_{i} ( \begin{matrix}x_{i} \\ y_{i} \end{matrix} ). Using the identity (easily proved using polar coordinates)

δ_{pq} p! = \frac{1}{π} ∫_{ℂ} x^{p}x̄^{q}e^{−|x|^{2}} dμ_{x} (p, q ∈ ℤ_{≥0}),

where dμ_{x} denotes the standard Lebesgue measure in ℂ, we obtain the integral representation

⟨x^{p}y^{q}, x^{p′}y^{q′}⟩ = \frac{1}{π^{2}} ∫_{ℂ^{2}}(−x)^{p}y^{q}ȳ^{p′}x̄^{q′}e^{−|x|^{2}−|y|^{2}} dμ_{x}dμ_{y}

or (changing x to −x) ⟨f_{1}, f_{2}⟩ = π^{−2} ∫_{ℂ^{2}} f_{1}(x, y)f_{2}(ȳ, −x̄)e^{−|x|^{2}−|y|^{2}}dμ_{x}dμ_{y} for the scalar product ⟨·, ·⟩ on ⨁_{a} V_{a}. This gives the integral representation

F (g; U, V) = \frac{1}{π^{2n}} ∫_{ℂ^{2n}} exp(− [∑_{i=1}^{n}(x_{i}x̄_{i} + y_{i}ȳ_{i}) − ∑_{i,j}(U_{ij}x_{i}^{∗}y_{j}^{∗} + V_{ij}x̄_{i}ȳ_{j})] ) dμ

for F (g; U, V), where dμ = dμ_{x_{1}} ⋯ dμ_{y_{n}} is the Lebesgue measure on ℂ^{2n}. Now this integral equals D^{−1/2} where D is the determinant of the quadratic form in square brackets, considered as a form of rank 4n over ℝ. The matrix representing this quadratic form with respect to the variables (y_{1}^{∗}, …, y_{n}^{∗}, −x_{1}^{∗}, …, −x_{n}^{∗}, ȳ_{1}, …, ȳ_{n}, −x̄_{1}, …, −x̄_{n}) is half of the one given in the proposition, where we have to remember that g^{t}( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} )g = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ). But we are taking the usual Lebesgue measure with respect to real parts and imaginary parts of x_{i} and y_{i} for the integral and we have ℜ(x) = (x + x̄)/2, ℑ(x) = (x − x̄)/2i, so the factors of 2 cancel. □

Theorem 8 in principle gives us a way to calculate the polynomials F_{μ μ μ,ν ν ν} for arbitrary n and for arbitrary μ μ μ and ννν, but these polynomials can be written uniquely in terms of the numbers t_{ij} = Tr(g_{i}\overline{g_{j}}) only if μ μ μ = ννν and n ≤ 4, because if μ μ μ ≠ ννν then they are only SO(4)- rather than O(4)-invariant, while if n > 4 then there are non-trivial relations among the t_{ij} (vanishing of all 5 × 5 minors) which mean that the representation of F_{ν ν ν}(g) as a polynomial in the t_{ij} is not unique. Therefore the most important cases of the theorem are n = 3 (which will be treated in detail in the next section) and n = 4 (where we have not been able to do any interesting calculations). However, using the results of §5 we can prove the following result for arbitrary values of n.

Proposition 2. The polynomials F_{ν ν ν} for d = 4 for any value of n coincide up to a scalar factor with P̃_{ν ν ν}^{M}, the pull-backs under β_{n} of the monomial basis polynomials P_{ν ν ν}^{M} ∈ P^{(n)}(4) .

Proof. Since both F_{ν ν ν} and P̃_{ν ν ν}^{M} are harmonic with respect to each variable g_{i} ∈ M_{2}(ℝ) = ℝ^{4}, it suffices by Proposition 2 of §6 to show that they agree up to a constant factor when all the g_{i} have determinant 0. In view of the definition of P_{ν ν ν}^{M}, this means that we must prove that

⟨gw_{ν ν ν}, w_{ν ν ν}⟩ = \frac{a!!!}{ννν!!!^{2}} ∏_{1≤i<j≤n}(g_{i}, g_{j})^{ν_{ij}}  if det(g_{1}) = ⋯ = det(g_{n}) = 0 ,  (64)

where we have now inserted the constant, using the abbreviations ννν!!! = ∏_{i<j} ν_{ij}!, a!!! = ∏_{i} a_{i}! .

To prove (64), we first observe that a 2 × 2 matrix of determinant 0 has rank ≤ 1 and hence can be written as ξξξ r^{t} for some column vectors ξξξ and r. So we can write each g_{i} as ξξξ_{i}r_{i}^{t} for some column vectors ξξξ_{i} = \binom{ξ_{i}}{η_{i}} and r_{i} = \binom{r_{i}}{s_{i}}. We first claim that the scalar product (g_{i}, g_{j}) is then given as the product of the two determinants |ξξξ_{i} ξξξ_{j}| and |r_{i} r_{j}|. This can be proved either by multiplying everything out and checking or, slightly more elegantly, by observing that

(g_{i}, g_{j}) = tr(g_{i}^{∗}g_{j}) = tr(−Jg_{i}^{t}Jg_{j}) = tr(−Jr_{i}ξξξ_{i}^{t}Jξξξ_{j}r_{j}^{t}) = (−ξξξ_{i}^{t}Jξξξ_{j}) tr(Jr_{i}r_{j}^{t}) = |ξξξ_{i} ξξξ_{j}| |r_{i} r_{j}| .

(Here J = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) as usual.) Next, we observe that g_{i}\binom{x_{i}}{y_{i}} = (r_{i}x_{i} + s_{i}y_{i})ξξξ_{i}, so we have

gw_{ν ν ν} = \frac{1}{ννν!!!} ∏_{i=1}^{n}(r_{i}x_{i} + s_{i}y_{i})^{a_{i}} · ∏_{1≤i<j≤n} |ξξξ_{i} ξξξ_{j}|^{ν_{ij}}

Finally, the definition of the G-invariant scalar product in V_{a} as defined in (63) is easily seen to be equivalent to

⟨(rx + sy)^{a}, P (x, y)⟩ = a! P (s, −r) for all P ∈ V_{a}.

Combining these three facts, we get the desired equality (64) :

⟨gw_{ν ν ν}, w_{ν ν ν}⟩ = \frac{1}{ννν!!!} ∏_{i<j} |ξξξ_{i} ξξξ_{j}|^{ν_{ij}} · ⟨∏_{i=1}^{n}(r_{i}x_{i} + s_{i}y_{i})^{a_{i}}, w_{ν ν ν}⟩

= \frac{a!!!}{ννν!!!^{2}} ∏_{i<j} (|ξξξ_{i} ξξξ_{j}| |r_{i} r_{j}|)^{ν_{ij}} = \frac{a!!!}{ννν!!!^{2}} ∏_{i<j} (g_{i}, g_{j})^{ν_{ij}} . □

Remark. The proof of Proposition 2 did not use the O(4)-invariance of F_{ν ν ν}, and hence gives another proof of this invariance, because of the second statement in Proposition 2 of §6.

§8. Higher spherical polynomials for n = 3

In the case n = 3, we have the extra fact that the polynomials F_{μ μ μ,ν ν ν} all vanish except those μ μ μ = ννν, since there is only one triple ννν with any given multidegree a, as given in (3), and the polynomials F_{μ μ μ,ν ν ν} with indices of different multidegrees vanish by definition. This case, which will be treated in detail in this section, is therefore particularly interesting.

It will be convenient to rename the coordinates of S_{n} for n = 3. We set

2 T = ( \begin{matrix}2m_{1} & r_{3} & r_{2} \\ r_{3} & 2m_{2} & r_{1} \\ r_{2} & r_{1} & 2m_{3} \end{matrix} ) .  (65)

We will use the notations m and r to denote the triples (m_{1}, m_{2}, m_{3}) and (r_{1}, r_{2}, r_{3}) and will consistently use the evident vector notation, e.g. m^{λ λ λ} for a triple λ λ λ = (λ_{1}, λ_{2}, λ_{3}) denotes ∏ m_{i}^{λ_{i}}.

The indices i are taken modulo 3. We write a typical element of N_{0}(a) as ( \begin{matrix}0 & ν_{3} & ν_{2} \\ ν_{3} & 0 & ν_{1} \\ ν_{2} & ν_{1} & 0 \end{matrix} ) with

ν_{i} ≥ 0, so that ν_{i} is the same as what was denoted ν_{i+1,i+2} in the last section. According to the result of §2, we know that P_{a}(d) is spanned by a unique (up to a scalar multiple) polynomial P_{ν ν ν}(T) = P_{ν ν ν}(m, r) whenever a and ννν are related as in (3). It is easily seen that P_{ν ν ν} is a homogeneous polynomial of tridegree ννν if we assign to r_{1}, r_{2}, r_{3}, m_{1}, m_{2}, and m_{3} the tridegrees (1,0,0), (0,1,0), (0,0,1), (−1,1,1), (1,−1,1), and (1,1,−1), respectively.

In terms of the coordinates (65), the differential operator D_{1} computed in §1 is given by

\frac{1}{4} D_{1} = (\frac{d}{2} + m_{1} \frac{∂}{∂m_{1}} + r_{2} \frac{∂}{∂r_{2}} + r_{3} \frac{∂}{∂r_{3}} )\frac{∂}{∂m_{1}} + m_{3} \frac{∂^{2}}{∂r_{2}^{2}} + r_{1} \frac{∂^{2}}{∂r_{2}∂r_{3}} + m_{2} \frac{∂^{2}}{∂r_{3}^{2}}  (66)

(and D_{2} and D_{3} by the same formula with the indices permuted cyclically).

This section, which is fairly long, will be divided into five subsections. In the first, we construct generating functions whose coefficients give us a canonical generator P_{ν ν ν}(T) of the 1-dimensional space P_{a}(d). In the second, we compute the relation between the new basis elements and the ones constructed in §5 and use this to compute the scalar products of the P_{ν ν ν} with themselves, while the third and fourth subsections contain various recursion relations and explicit formulas for the coefficients of these polynomials. The last subsection contains a brief discussion of the n = 3 case of the inhomogeneous coordinates defined at the end of §1, of the related angular coordinates, and of a somewhat surprising extra symmetry of these coordinates.

A. Generating function. We begin by writing out the generating function of Proposition 2 explicitly for n = 3. This generating function will then be generalized to arbitrary d.

For the case d = 4, we can choose P_{ν ν ν}(T) canonically as F_{ν ν ν}(g) = ⟨gw_{ν ν ν}, w_{ν ν ν}⟩ defined as in §6, where g = (g_{1}, g_{2}, g_{3}) is a triple of matrices in M_{2}(ℝ) related to the 3 × 3 matrix T by m_{i} = det(g_{i}), r_{i} = Tr(g_{i+1}g_{i+2}^{∗}). We introduce dummy variables X_{1}, X_{2}, X_{3} and define X^{ν ν ν} as usual as X_{1}^{ν_{1}}X_{2}^{ν_{2}}X_{3}^{ν_{3}}.

Proposition 1. The polynomials P_{ν ν ν}(T) for n = 3, d = 4 are given by the generating function

∑_{ν ν ν} P_{ν ν ν}(T) X^{ν ν ν} = \frac{1}{\sqrt{Δ_{0}(T, X)^{2} − 4d(T)X_{1}X_{2}X_{3}}} ,  (67)

where

Δ_{0}(T, X) = 1 − ∑_{i=1}^{3}(r_{i}X_{i} − m_{i}r_{i}X_{i+1}X_{i+2} − m_{i+1}m_{i+2}X_{i}^{2})

and

d(T) = 4 det(T) = ∏_{i=1}^{3} r_{i} − ∑_{i=1}^{3} m_{i}r_{i}^{2} + 4 ∏_{i=1}^{3} m_{i} .

Proof. This is just Theorem 8 for the special case n = 3, since we have already seen that in that case the terms with μ μ μ ≠ ννν in the generating function vanish. To compute the generating function explicitly we must calculate the 12 × 12 determinant D(g; U, V) defined in Theorem 8. By direct calculation we find D(g; U, V) = Δ_{0}(T, X)^{2} − 4d(T)X_{1}X_{2}X_{3} when all m_{i} = 1, where T is determined by g as explained above and X_{i} = U_{i+1,i+2}V_{i+1,i+2}, and the general case then follows by homogeneity. □

Proposition 1 makes it easy to compute any polynomial P_{ν ν ν} explicitly. Here is a table for all ννν with |ννν| ≤ 4, where we give only one representative for each 𝔖_{3}-orbit of indices ννν:

P_{000} = 1 ;  P_{100} = r_{1} ;  P_{200} = r_{1}^{2} − m_{2}m_{3} , P_{110} = 2r_{1}r_{2} − m_{3}r_{3} ;
P_{300} = r_{1}^{3} − 2m_{2}m_{3}r_{1} ,  P_{210} = 3r_{1}^{2}r_{2} − 2m_{3}(r_{1}r_{3} + m_{2}r_{2}) ,
P_{111} = 8r_{1}r_{2}r_{3} − 4(m_{1}r_{1}^{2} + m_{2}r_{2}^{2} + m_{3}r_{3}^{2}) + 8m_{1}m_{2}m_{3} ;
P_{400} = r_{1}^{4} − 3m_{2}m_{3}r_{1}^{2} + m_{2}^{2}m_{3}^{2} ,  P_{310} = 4r_{1}^{3}r_{2} − 3m_{3}r_{1}^{2}r_{3} − 6m_{2}m_{3}r_{1}r_{2} + 2m_{2}m_{3}^{2}r_{3} ,
P_{220} = 6r_{1}^{2}r_{2}^{2} − 6m_{3}r_{1}r_{2}r_{3} − 3m_{3}(m_{1}r_{1}^{2} + m_{2}r_{2}^{2}) + m_{3}^{2}r_{3}^{2} + 2m_{1}m_{2}m_{3}^{2} ,
P_{211} = 18r_{1}^{2}r_{2}r_{3} − 9m_{1}r_{1}^{3} − 12r_{1}(m_{2}r_{2}^{2} + m_{3}r_{3}^{2}) − 4m_{2}m_{3}r_{2}r_{3} + 26m_{1}m_{2}m_{3}r_{1} .

We now state a generalization of Proposition 1 which gives explicit higher spherical polynomials for n = 3 and for arbitrary values of ννν and d as the coefficients of a generating function.

Theorem 9. For T ∈ S_{3} as in (65) and X = (X_{1}, X_{2}, X_{3}) let Δ_{0}(T, X) and d(T) be as in Proposition 1 and set

R(T, X) = \frac{Δ_{0}(T, X) + \sqrt{Δ_{0}(T, X)^{2} − 4d(T)X_{1}X_{2}X_{3}}}{2} .  (68)

For any d ∈ ℂ, define polynomials P_{ν ν ν,d}(T) (ννν = (ν_{1}, ν_{2}, ν_{3}) ∈ ℤ_{≥0}^{3}) by the generating function

∑_{ν ν ν} P_{ν ν ν,d}(T) X^{ν ν ν} = \frac{R(T, X)^{−s}}{\sqrt{Δ_{0}(T, X)^{2} − 4d(T)X_{1}X_{2}X_{3}}} .  (69)

where s = d/2 − 2. Then P_{ν ν ν,d}(T) belongs to the space P_{a}(d), where a is related to ννν by (3), and generates this space if d ∉ {2, 0, −2, −4, … } or if d = 2 and all ν_{i} are strictly positive.

Proof. Using the formula for D_{1} given in (66), we find by direct calculation that

\frac{1}{4m_{2}m_{3} − r_{1}^{2}} D_{1}(\frac{d(T)^{n}}{Δ_{0}(T, X)^{2n+s+1}} )

= n(n + s) \frac{d(T)^{n−1}}{Δ_{0}(T, X)^{2n+s+1}} − (2n + s + 1)(2n + s + 2) \frac{d(T)^{n} X_{1}X_{2}X_{3}}{Δ_{0}(T, X)^{2n+s+3}} .

Applying this identity to the expansion

\frac{1}{\sqrt{Δ_{0}^{2} − 4d(T)X}} (\frac{Δ_{0} + \sqrt{Δ_{0}^{2} − 4d(T)X}}{2} )^{−s} = ∑_{n=0}^{∞} \binom{2n + s}{n} \frac{(d(T)X)^{n}}{Δ_{0}^{2n+s+1}}  (70)

shows that D_{1}(P_{ν ν ν,d}) = 0. The vanishing of the other D_{i} follows in the same way or by symmetry. The final statement of the theorem follows because dim P_{a}(d) = 1 for d ∉ 2ℤ_{≤0} and since P_{ν ν ν,d} ≢ 0 for −s ∉ ℕ by the calculation of its leading term as given in Proposition 2 below. □

Using Theorem 9, we can compute the polynomials P_{ν ν ν,d} of any given degree, a typical value (generalizing to arbitrary s the one given above for s = 0) being

P_{211,d}(T) = \frac{1}{2}(s + 2)^{2}(s + 3)^{2}r_{1}^{2}r_{2}r_{3} − \frac{1}{2}(s + 2)(s + 3)^{2}m_{1}r_{1}^{3} − (s + 2)^{2}(s + 3)(m_{2}r_{2}^{2} + m_{3}r_{3}^{2})r_{1}
− (s + 1)(s + 2)^{2}m_{2}m_{3}r_{2}r_{3} + (s + 2)(5s + 13)m_{1}m_{2}m_{3}r_{1} .

The polynomials P_{ν ν ν,d}(T) for all ννν with |ννν| ≤ 6 are given in Table 1 at the end of the paper.

Remark 1. The expressions Δ_{0}(T, X) and d(T)X_{1}X_{2}X_{3} appearing in the theorem can be written as (1 − σ_{1}/2)^{2} − σ_{2} and 2σ_{3}, respectively, where the σ_{i} are the elementary symmetric polynomials in the eigenvalues (= coefficients of the characteristic polynomial up to sign) of the

3 × 3 matrix ( \begin{matrix}0 & X_{3} & X_{2} \\ X_{3} & 0 & X_{1} \\ X_{2} & X_{1} & 0 \end{matrix} )T . This remark, and the generating function of Theorem 8, will be

greatly generalized in §9.

Remark 2. In the above proof, we simply wrote down the right-hand side of (69) as a generalization of (67) and verified that its coefficients were spherical polynomials, which was an easy calculation. The main point here is how to guess the correct formula (69). This was done (roughly) in the following way. We wish to define for all d a generator P_{ν ν ν,d} of the 1-dimensional space P_{a}(d) in such a way that the sum ∑_{ν ν ν} P_{ν ν ν,d}(T)X^{ν ν ν} is algebraic and agrees with (67) if d = 4. Because of the one-dimensionality, we know that these functions, if they exist, must be scalar multiples of both the monomial basis elements P_{ν ν ν}^{M} and the descending basis elements P_{ν ν ν}^{D} as defined in §5, where the multiples are determined by computing the constant term and the effect of the mixed Laplace operators D_{ij}, respectively. So we begin by computing these things in the already known case d = 4 (cf. Subsection B below, where these computations are done, assuming the definition (69), for all d), and then insert s into the formulas obtained in the simplest way possible such that the coefficients of P_{ν ν ν} do not acquire any denominators (which are prohibited if ∑ P_{ν ν ν}X^{ν ν ν} is to be an algebraic function). This led to formulas (74) and (77) below, after which the generating function (69) was found more or less by doing the calculations of Subsection B in the reverse order.

Remark 3. It is interesting to compare the computations for n = 3 in the last two sections (Proposition 1 and Theorem 9) with the classical case n = 2. In this case the polynomial F_{μ μ μ,ν ν ν}(g) defined in §7 equals P_{a}(T) if μ μ μ = ννν = ( \begin{matrix}0 & a \\ a & 0 \end{matrix} ) for some integer a ≥ 0 and is 0 otherwise, where

T = ( \begin{matrix}m_{1} & r/2 \\ r/2 & m_{2} \end{matrix} ) with m_{i} = det(g_{1}) and r = tr(g_{1}g_{2}^{∗}). The 8 × 8 determinant D(g; U, V) occurring

in Theorem 8, for U = ( \begin{matrix}0 & −u \\ u & 0 \end{matrix} ) and V = ( \begin{matrix}0 & −v \\ v & 0 \end{matrix} ), is easily calculated to be (X^{2} − rX + m_{1}m_{2})^{2}, where X = uv, so we obtain the generating function ∑_{a=0}^{∞} P_{a}(T) X^{a} = (1 − rX + m_{1}m_{2}X^{2})^{−1} for the spherical polynomials P_{a}(T) when d = 4. On the other hand, in Example 1 of §1 we calculated a generator of the one-dimensional space P_{(a,a)}(d) for any a ≥ 0 and any d ∉ 2ℤ ∩ [4 − 2a, 2 − a], obtaining (with a suitable normalization) the formula

P_{a,d}( \begin{matrix}m_{1} & r/2 \\ r/2 & m_{2} \end{matrix} ) = ∑_{0≤l≤a/2} \binom{a − l}{l}\binom{a − l + d/2 − 2}{a − l} (−m_{1}m_{2})^{l} r^{a−2l} ,  (71)

which in turn is equivalent to the more general generating function

∑_{a=0}^{∞} P_{a,d}( \begin{matrix}m_{1} & r/2 \\ r/2 & m_{2} \end{matrix} ) X^{a} = (1 − rX + m_{1}m_{2}X^{2})^{1−d/2} .  (72)

The formulas (71) and (72) are of course classical, the polynomials P_{a,d} being called Gegenbauer polynomials in that case (with the special names Legendre polynomials if d = 3 and Chebyshev polynomials if d = 2 or 4). In §9 we will show how to produce natural generating functions generalizing (72) and (69) that give us canonical bases of P_{a}(d) for arbitrary values of n.

Notation. Let us fix the notation for the coefficients of the polynomials P_{ν ν ν} = P_{ν ν ν,d}. We write T as in equation (65) and denote by C(λ λ λ;μ μ μ) the coefficient of m^{λ λ λ}r^{μ μ μ} in P_{ν ν ν}. Clearly this coefficient is 0 unless ννν, λ λ λ and μ μ μ are related by

ν_{i} = μ_{i} − λ_{i} + λ_{i+1} + λ_{i+2} ,  λ_{i} = \frac{1}{2}(ν_{i+1} + ν_{i+2} − μ_{i+1} − μ_{i+2})  (73)

(here i is considered modulo 3), so that we can omit the subscript ννν in C(λ λ λ;μ μ μ), but we can also write C_{ν ν ν}(λ λ λ;μ μ μ) if this is needed for emphasis or clarity. We will also almost always omit the “d” from the notation for the coefficients of P_{ν ν ν,d}, but of course these coefficients depend (polynomially) on this parameter. As usual, we also work with s, where d = 2s + 4.

B. Ratios of the various basis elements. In Chapter I, we gave two special bases {P_{ν ν ν}^{M}} and {P_{ν ν ν}^{D}} for P_{n}(d). In the special case n = 3, we have now given a third basis {P_{ν ν ν}} from the generating function (69). Since the (non-zero) spaces P_{a}(d) for n = 3 are all onedimensional, all three bases agree up to constants. In this subsection we compute these constants of proportionality and at the same time the constant terms and the norms (with respect to the canonical scalar product) of the polynomials P_{ν ν ν}.

Proposition 2. The higher spherical polynomial P_{ν ν ν}(T) defined by the generating function (69) for n = 3 is related to the monomial basis by

P_{ν ν ν}(T) = 2^{ν_{1}+ν_{2}+ν_{3}} \frac{(ν_{1} + ν_{2} + s)! (ν_{1} + ν_{3} + s)! (ν_{2} + ν_{3} + s)!}{ν_{1}! ν_{2}! ν_{3}! (ν_{1} + s)! (ν_{2} + s)! (ν_{3} + s)!} P_{ν ν ν}^{M}(T) .  (74)

Proof. In view of the definition of the monomial basis, this is equivalent to the statement that the leading coefficient of the polynomial P_{ν ν ν,d} is given by

C(000;ννν) = \binom{ν_{1} + ν_{2} + s}{ν_{1}}\binom{ν_{2} + ν_{3} + s}{ν_{2}}\binom{ν_{3} + ν_{1} + s}{ν_{3}} .  (75)

Taking T with m_{1} = m_{2} = m_{3} = 0 in (69) and (70), so that P_{ν ν ν}(T) = C(000;ννν) r^{ν ν ν}, and setting t_{i} = r_{i}X_{i}, we find

∑_{ν ν ν} C(000;ννν) t^{ν ν ν} = ∑_{n=0}^{∞} \binom{2n + s}{n} \frac{(t_{1}t_{2}t_{3})^{n}}{(1 − t_{1} − t_{2} − t_{3})^{2n+s+1}}

= ∑_{n,e≥0} \binom{2n + s}{n}\binom{2n + s + e}{e} (t_{1}t_{2}t_{3})^{n} (t_{1} + t_{2} + t_{3})^{e}

= ∑_{n,a,b,c≥0} \frac{(2n + s + a + b + c)!}{n! (n + s)! a! b! c!}t_{1}^{a+n}t_{2}^{b+n}t_{3}^{c+n} .

Hence

C(000;ννν) = ∑_{n=0}^{min(ν_{1},ν_{2},ν_{3})} \frac{(ν_{1} + ν_{2} + ν_{3} + s − n)!}{n! (n + s)! (ν_{1} − n)! (ν_{2} − n)! (ν_{3} − n)!} .  (76)

This expression is symmetric in the three arguments ν_{i}, but more complicated than the formula given in the proposition. To prove that formula, we first break the symmetry. We rewrite (76) as C(000;ννν) = \binom{ν_{1}+ν_{2}+s}{ν_{1}} C′(ννν) with C′(ννν) defined by

C′(ννν) = ∑_{n=0}^{min(ν_{1},ν_{2},ν_{3})} \binom{ν_{1}}{n}\binom{ν_{2} + s}{ν_{2} − n}\binom{ν_{1} + ν_{2} + ν_{3} + s − n}{ν_{3} − n} .

Then for fixed ν_{1} we find the generating function

∑_{ν_{2}, ν_{3}≥0}C′(ν_{1}, ν_{2}, ν_{3}) X^{ν_{2}} Y^{ν_{3}}

= ∑_{n=0}^{ν_{1}} \binom{ν_{1}}{n} ∑_{ν_{2}=n}^{∞} \binom{ν_{2} + s}{ν_{2} − n}X^{ν_{2}} ∑_{ν_{3}=n}^{∞} \binom{ν_{1} + ν_{2} + ν_{3} + s − n}{ν_{3} − n} Y^{ν_{3}}

= ∑_{n=0}^{ν_{1}} \binom{ν_{1}}{n} ∑_{ν_{2}=n}^{∞} \binom{ν_{2} + s}{ν_{2} − n}\frac{X^{ν_{2}} Y^{n}}{(1 − Y)^{ν_{1}+ν_{2}+s+1}}

= ∑_{n=0}^{ν_{1}} \binom{ν_{1}}{n}\frac{X^{n} Y^{n}}{(1 − Y)^{ν_{1}+n+s+1}} \frac{1}{(1 − X/(1 − Y))^{n+s+1}}

= \frac{1}{(1 − Y)^{ν_{1}} (1 − X − Y)^{s+1}} (1 + \frac{XY}{1 − X − Y} )^{ν_{1}}

= \frac{(1 − X)^{ν_{1}}}{(1 − X − Y)^{ν_{1}+s+1}} = ∑_{ν_{3}=0}^{∞} \binom{ν_{1} + s + ν_{3}}{ν_{3}} \frac{Y^{ν_{3}}}{(1 − X)^{ν_{3}+s+1}}

= ∑_{ν_{2}=0}^{∞} ∑_{ν_{3}=0}^{∞} \binom{ν_{1} + ν_{3} + s}{ν_{3}}\binom{ν_{2} + ν_{3} + s}{ν_{2}}X^{ν_{2}} Y^{ν_{3}}

proving (75). □

We now give the relationship between P_{ν ν ν} and P_{ν ν ν}^{D}. The following proposition is just the special case n = 3 of equation (135) of §9 below, which says that for any n the higher spherical polynomials defined by a generalization (117) of the generating series (69) are proportional to the descending basis elements, with a proportionality factor depending only on the total weight k = \frac{1}{2} a · 1 . Recall the Pochhammer symbol (x)_{k} = x(x + 1) ⋯ (x + k − 1).

Proposition 3. The polynomial P_{ν ν ν}(T) is related to the descending basis by

P_{ν ν ν}(T) = 2^{k} (s + 2)_{k} (2s + 2)_{k} P_{ν ν ν}^{D}(T) ,  (77)

where k = ν_{1} + ν_{2} + ν_{3}.

Now combining Proposition 2 and 3 with Theorem 6, we deduce the formula for the scalar product of P_{ν} ∈ P_{a}(d) with itself:

Corollary. For every ννν = (ν_{1}, ν_{2}, ν_{3}) ∈ ℤ_{≥0}^{3} we have

(P_{ν ν ν}, P_{ν ν ν})_{d} = 2^{2k} \binom{ν_{1} + ν_{2} + s}{ν_{1}}\binom{ν_{2} + ν_{3} + s}{ν_{2}}\binom{ν_{3} + ν_{1} + s}{ν_{3}} (s + 2)_{k} (2s + 2)_{k} , (78)

where d = 2s + 4 and k = ν_{1} + ν_{2} + ν_{3}. □

Equation (78) can be used to simplify the computation of one of the coefficients in the recursion (82) in Proposition 5 below. Conversely, as explained in the proof of that proposition, one can also obtain the coefficient in question by direct computation and then derive (78) from it, after which one can derive Proposition 3 using Theorem 6 and the fact that all of our bases are a priori proportional, without needing to rely on the considerably harder results from §9.

C. Recursion relations. In Subsection A we defined canonical polynomials P_{ν ν ν} ∈ P^{(3)}(d) by means of a generating function. We now give five different recursion relations for the coefficients of these polynomials, coming from the differential equation, the scalar product, the raising operators, the mixed Laplacians, and the generating function, respectively.

1. Recursion relation coming from the differential equation. The first recursion is for the coefficients of a single polynomial P_{ν ν ν}. According to Theorem 1 and its proof, all of the coefficients of P_{ν ν ν} are determined (in fact, overdetermined) recursively by the harmonicity and homogeneity properties from the “constant term” (specialization to t_{11} = t_{22} = t_{33} = 0, or to m_{1} = m_{2} = m_{3} = 0 in the notation (65)). We make this explicit. In Subsection A we gave the formula for the differential operator D_{1} in terms of the new coordinates (65), and of course the formulas for the other D_{i} are obtained by cyclic permutation of the indices. The equation D_{1}P_{ν ν ν} = 0 gives the recursion (with s = d/2 − 2 as before)

λ_{1}(λ_{1} + μ_{2} + μ_{3} + s + 1) C(λ_{1}, λ_{2}, λ_{3}; μ_{1}, μ_{2}, μ_{3})
+ (μ_{2} + 1)(μ_{2} + 2) C(λ_{1} − 1, λ_{2}, λ_{3} − 1; μ_{1}, μ_{2} + 2, μ_{3})
+ (μ_{2} + 1)(μ_{3} + 1) C(λ_{1} − 1, λ_{2}, λ_{3}; μ_{1} − 1, μ_{2} + 1, μ_{3} + 1)
+ (μ_{3} + 1)(μ_{3} + 2) C(λ_{1} − 1, λ_{2} − 1, λ_{3}; μ_{1}, μ_{2}, μ_{3} + 2) = 0 .  (79)

This gives any C(λ λ λ;μ μ μ) in terms of C(λ λ λ′;μ μ μ′) with λ′_{1} < λ_{1} and hence by induction reduces everything to the case λ_{1} = 0. Now the recurrence given by D_{2}P_{ν ν ν} = 0 gives

λ_{2}(λ_{2} + μ_{1} + μ_{3} + s + 1) C(0, λ_{2}, λ_{3}; μ_{1}, μ_{2}, μ_{3})
+ (μ_{1} + 1)(μ_{1} + 2) C(0, λ_{2} − 1, λ_{3} − 1; μ_{1} + 2, μ_{2}, μ_{3})
+ (μ_{1} + 1)(μ_{3} + 1) C(0, λ_{2} − 1, λ_{3}; μ_{1} + 1, μ_{2} − 1, μ_{3} + 1) = 0 ,  (80)

which lets us get down to λ_{2} = 0, and similarly the recurrence from D_{3} gives

λ_{3}(λ_{3} + μ_{1} + μ_{2} + s + 1) C(0, 0, λ_{3}; μ_{1}, μ_{2}, μ_{3})
+ (μ_{1} + 1)(μ_{2} + 1) C(0, 0, λ_{3} − 1; μ_{1} + 1, μ_{2} + 1, μ_{3} − 1) = 0 ,  (81)

which gets us down to λ λ λ = (0, 0, 0). But for fixed ννν there is only one coefficient C(λ λ λ;μ μ μ) with λ λ λ = (0, 0, 0), namely C(000;ννν), which is given by Proposition 2 above. (This is the only point where our specific normalization of the basis element of P_{a}(d) for n = 3 is used.) Summarizing, we have:

Proposition 4. The coefficients of C(λ λ λ;μ μ μ) are determined by the recursive formulas (79)–(81) together with the initial conditions (75).

We note, however, that determining the coefficients explicitly from the recursions (79)–(81) is quite difficult. We will discuss this problem, and solve it in some cases, in Subsection D.

2. Recursion relation coming from the scalar product. In the method just described, we fixed the index ννν and gave a recursive relation for the coefficients C(λ λ λ;μ μ μ) = C_{ν ν ν}(λ λ λ;μ μ μ) of P_{ν ν ν}(T) by using the differential equation which this polynomial satisfies. A different approach is to give a recursive formula for the polynomials P_{ν ν ν} themselves (and hence implicitly also for their coefficients), as follows:

Proposition 5. For all ν_{1}, ν_{2}, ν_{3} ≥ 0 we have

\begin{aligned}ν_{1}(ν_{1} + s) P_{ν_{1},ν_{2},ν_{3}} = & (ν_{1} + ν_{2} + s)(ν_{1} + ν_{3} + s) r_{1} P_{ν_{1}−1,ν_{2},ν_{3}} \\ & − (ν_{2} + 1)(ν_{2} + s + 1) m_{2} P_{ν_{1}−1,ν_{2}+1,ν_{3}−1} \\ & − (ν_{3} + 1)(ν_{3} + s + 1) m_{3} P_{ν_{1}−1,ν_{2}−1,ν_{3}+1} \\ & − (ν_{1} + ν_{2} + ν_{3} + s)(ν_{1} + ν_{2} + ν_{3} + 2s) m_{2}m_{3} P_{ν_{1}−2,ν_{2},ν_{3}} \end{aligned} (82)

with the convention that any term having a negative index is to be interpreted as zero.

Proof. The idea is similar to the classical proof of the 3-term recursion for orthogonal polynomials {P_{ν}(x)} in one variable (corresponding to n = 2 in our setup). There one observes that, since the polynomial P_{ν}(x) has degree exactly ν, one can write xP_{ν−1}(x) as a linear combination of P_{ν−i}(x) (i = 0, 1, …), and that the coefficient of P_{ν−i} in this combination vanishes for i ≥ 3 because then (xP_{ν−1}, P_{ν−i}) = (P_{ν−1}, xP_{ν−i}) = 0 by orthogonality. This gives xP_{ν−1} = a_{ν}P_{ν} + b_{ν}P_{ν−1} + c_{ν}P_{ν−2}. The coefficient a_{ν} can be computed by comparing the coefficients of x^{ν} on both sides. The coefficient b_{ν} can be computed in a similar way, but by a longer calculation, if one knows the subleading coefficients of the polynomials, and the coefficient c_{ν} by a similar but even longer calculation if one knows the third coefficients from the top. Alternatively, the coefficients c_{ν} can be obtained from the calculation

c_{ν+1} (P_{ν−1}, P_{ν−1}) = (xP_{ν}, P_{ν−1}) = (P_{ν}, xP_{ν−1}) = a_{ν}(P_{ν}, P_{ν})

if one knows the norms of the polynomials P_{ν}, or alternatively one can turn this calculation around to obtain a recursive formula for these norms if one knows the c_{ν}. Here the same arguments apply. Write the product r_{1}P_{ν−1,ν_{2},ν_{3}}(T) as a linear combination of other higher spherical polynomials P_{ν ν ν}′(T), where we can use the homogeneity of the polynomials to simplify the formulas by setting m_{1} = m_{2} = m_{3} = 1. Then the same scalar product argument as before gives that the coefficients of P_{ν ν ν}′ in this linear combination vanish for all multi-indices ννν′ except those occurring in (82). The coefficient of P_{ν_{1},ν_{2},ν_{3}} in (82) is found easily by comparing the coefficients of r_{1}^{ν_{1}}r_{2}^{ν_{2}}r_{3}^{ν_{3}} on both sides of the equation, using formula (75). The coefficient of P_{ν_{1}−1,ν_{2}+1,ν_{3}−1} is only slightly harder, because its top monomial r_{1}^{ν_{1}−1}r_{2}^{ν_{2}+1}r_{3}^{ν_{3}−1} does not occur in any of the other terms on the right-hand side of (82) and the coefficient of this monomial in P_{ν_{1},ν_{2},ν_{3}} (where it corresponds to λ λ λ = (0, 1, 0)) can be obtained easily from (75) and the recursion (81) or as a special case of Proposition 8 below. The calculation for P_{ν_{1}−1,ν_{2}−1,ν_{3}+1} is of course exactly similar. Finally, the coefficient of P_{ν_{1}−2,ν_{2},ν_{3}} can be obtained in a similar way, though with a little more computation, using the recursion (80) above or Proposition 9 in Subsection D below. (Alternatively, they can be obtained with less computation by computing the scalar products of both sides of (82) with P_{ν_{1}−2,ν_{2},ν_{3}} and using the orthogonality of the higher spherical polynomials together with the explicit formula (78) for their norms. Conversely, as already mentioned at the end of Subsection B, we can use the direct computation to obtain alternative proofs of (78) and then of (77) that do not require appealing to the more general results in §9.) The calculations are straightforward, though lengthy, and will be omitted. □

As well as (82) one of course also has the corresponding formulas for any cyclic permutation of the indices i of ν_{i}, m_{i}, and r_{i}, since P_{ν ν ν}(T) is symmetric under these permutations. This result lets one compute all of these polynomials recursively starting with the initial value P_{0,0,0} = 1: equation (81) lets one successively reduce the value of ν_{1} by (at least) one and hence expresses any P_{ν_{1},ν_{2},ν_{3}} in terms of polynomials P_{0,ν′_{2},ν′_{3}}, and then applying equation (81) with the indices cyclically permuted one reduces in turn to polynomials of the form P_{0,0,ν′′_{3}} and finally to P_{0,0,0}.

3. Recursion relation coming from the raising operators. The recursion just given expresses the polynomial P_{ν ν ν} as a linear combination of four polynomials P_{ν ν ν}′ with “smaller” values of ννν′. The raising operators introduced in §5 permit us to do even better in the sense that we can express P_{ν ν ν} in terms of a single predecessor (which can be chosen at will to be P_{ν_{1}−1,ν_{2},ν_{3}}, P_{ν_{1},ν_{2}−1,ν_{3}} or P_{ν_{1},ν_{2},ν_{3}−1}), but at the expense of using second order differential operators. More concretely, equation (51) gives us the relation

4(ν_{1} + ν_{2} + s)(ν_{1} + ν_{3} + s) P_{ν_{1},ν_{2},ν_{3}}^{M} = R_{23}(P_{ν_{1}−1,ν_{2},ν_{3}}^{M}) ,

where R_{23} is the operator defined in (47) (and of course similar formulas with R_{13} and R_{12}), and then we can use (74) to reexpress this identity in terms of our preferred basis {P_{ν ν ν}} as

2ν_{1}(ν_{1} + s) P_{ν_{1},ν_{2},ν_{3}} = R_{23}(P_{ν_{1}−1,ν_{2},ν_{3}}) .  (83)

Notice that this recursion, unlike the others discussed here, works equally well for all values of n, not just for n = 3.

4. Recursion relation coming from the mixed Laplacians. A fourth approach is to use the defining property (55) of the descending basis elements P_{ν ν ν}^{D}, together with the relation (77) between these functions and our preferred basis elements P_{ν ν ν}. We do not write this out in detail since it is not particularly illuminating.

5. Recursion relation coming from the generating function. Finally, we can try to compute all of the coefficients of the P_{ν ν ν}(T) from the generating function (69), rather than only the constant terms as was done in Proposition 2. A closed formula seems very hard to obtain in this way, but we can obtain yet one more recursive formula for these coefficients, of a somewhat peculiar type, as follows. The quantity R = R(T, X) defined in (68) satisfies the quadratic equation R^{2} − Δ_{0}(T, X)R + d(T)X_{1}X_{2}X_{3} = 0, where Δ_{0}(T, X) and d(T) are given in Proposition 1, so from (69) we deduce the generating function identity

∑_{ν ν ν} P_{ν ν ν,d}(T) X^{ν ν ν} − Δ_{0}(T, X) ∑_{ν ν ν} P_{ν ν ν,d+2}(T) X^{ν ν ν} + d(T)X_{1}X_{2}X_{3} ∑_{ν ν ν} P_{ν ν ν,d+4}(T) X^{ν ν ν} = 0 ,

and by comparing coefficients we obtain from this a formula for P_{ν ν ν,d} as a linear combination of 11 values of P_{ν ν ν′,d+2} and P_{ν ν ν′,d+4} .

D. Explicit formulas for the coefficients of P_{ν ν ν,d}. In this subsection we compute as far as we are able the coefficients of the polynomials P_{ν ν ν,d}. The calculations require, and reveal, some quite surprising combinatorial identities. However, the formulas are rather complicated and are not used again, so that the reader who does not like this sort of thing can skip to Susbsection E without any loss of continuity. We use the recursions given in the first paragraph of Subsection B, i.e., we use (81), (80) and (79) in succession. The first step is easy, since from formulas (81) and (75) we immediately find by induction the formula for C(λ λ λ;μ μ μ) when two of the λ_{i} vanish:

Proposition 6. Let λ λ λ = (0, 0, λ_{3}) and μ μ μ = (μ_{1}, μ_{2}, μ_{3}) ≥ λ λ λ. Then

C(λ λ λ;μ μ μ) = \frac{(−1)^{λ_{3}}(μ_{1} + μ_{2} + λ_{3} + s)!(μ_{1} + μ_{3} + s)!(μ_{2} + μ_{3} + s)!}{λ_{3}!μ_{1}!μ_{2}!(μ_{3} − λ_{3})!(μ_{1} + λ_{3} + s)!(μ_{2} + λ_{3} + s)!(μ_{3} − λ_{3} + s)!} .

Note that this is again a simple closed multiplicative formula, whereas using the generating function directly would have given a formula for C(0, 0, λ_{3};μ μ μ) as a double sum.

We next consider the C(λ λ λ;μ μ μ) where only one of the λ_{i} vanishes. Here the formula which we will obtain is no longer multiplicative, but it is a simple sum of products of binomial coefficients rather than the four-fold sum that we would get if we simply expanded the generating function (69).

Equation (80) is equivalent by induction over λ_{2} to the formula

C(0, λ_{2}, λ_{3};μ μ μ) =\frac{(−1)^{λ_{2}} (μ_{1} + μ_{3} + λ_{2} + s)!}{μ_{1}! μ_{3}! (μ_{1} + μ_{3} + 2λ_{2} + s)!} ∑_{n=max(0,λ_{2}−λ_{3})}^{min(λ_{2},μ_{2})} \frac{(μ_{3} + n)!(μ_{1} + 2λ_{2} − n)!}{n! (λ_{2} − n)!}
× C(0, 0, λ_{3} − λ_{2} + n; μ_{1} + 2λ_{2} − n, μ_{2} − n, μ_{3} + n) .

Substituting into this the formula from Proposition 6, we find the expression

C(0, λ_{2}, λ_{3};μ μ μ) = = (−1)^{λ_{3}} \frac{(μ_{2} + μ_{3} + s)! (μ_{1} + μ_{2} + λ_{3} + s)!(μ_{1} + μ_{3} + λ_{2} + s)!}{μ_{1}! (ν_{2} + s)! (ν_{3} + s)!}
× S(−μ_{3}; 0, λ_{2} − λ_{3}; λ_{2}, μ_{2}; μ_{1} + μ_{2} + λ_{2} + λ_{3} + s),  (84)

where S(a; b, c; d, e; f) for integers a ≤ b, c ≤ d, e ≤ f is defined by

S(a; b, c; d, e; f) = \frac{1}{(b − a)!(c − a)!(f − d)!(f − e)!} ∑_{n=max(b,c)}^{min(d,e)} \frac{(−1)^{n} (n − a)! (f − n)!}{(n − b)! (n − c)! (d − n)! (e − n)!}.

Formula (84) expresses C(λ λ λ;μ μ μ) when λ_{1} = 0 as a simple sum of multinomial coefficients as opposed to the quadruple sum expression which would have been obtained by using the generating function (69). On the other hand, it is not symmetric in the indices “2” and “3”. In looking for a symmetric expression, we discover the remarkable symmetry property

(−1)^{d+e}S(a; b, c; d, e; f) = S(c; d, e; f, d + e − b; f − a + c)  (85)

(where the right-hand side is zero if f − a < d + e − b − c). The proof is amusing and the reader may enjoy looking for it. Applying (85) to (84) gives the new expression

\begin{matrix}C(0, λ_{2}, λ_{3}; & μ μ μ) = \frac{(μ_{2} + μ_{3} + s)!}{μ_{1}! λ_{2}! λ_{3}! (ν_{2} + s)! (ν_{3} + s)!} \\ × ∑_{n=max(λ_{2},λ_{3})}^{min(μ_{3}+λ_{2},μ_{2}+λ_{3})} \frac{(−1)^{n} n! (ν_{1} + ν_{2} + ν_{3} + s − n)!}{(n − λ_{2})! (n − λ_{2})! (μ_{3} + λ_{2} − n)! (μ_{2} + λ_{3} − n)!} , \end{matrix}  (86)

which is again a simple sum of multinomial coefficients but is now symmetric in indices 2 and 3.

We now look for an explanation of the symmetry property (85). Combining (85) with the trivial symmetries under b ↔ c and d ↔ e, (a; b, c; d, e; f) ↦ (a+k; b+k, c+k; d+k, e+k; f +k) (k ∈ ℤ, and up to sign (−1)^{k}) and (a; b, c; d, e; f) ↦ (k − f; k − e, k − d; k − c, k − b; k − a) (again up to sign), we find that there are 9 essentially different sums ±S which are equal by (85). The nine inequalities on the variables a through f can be made uniform by writing (a; b, c; d, e; f) as (a_{1} + a_{2} − b_{3}; a_{1}, a_{2}; b_{1}, b_{2}; b_{1} + b_{2} − a_{3}) where b_{i} ≥ a_{j} for all i and j. The formula which makes all symmetries evident is then

\begin{matrix}S( & a_{1} + a_{2} − b_{3}; a_{1}, a_{2}; b_{1}, b_{2}; b_{1} + b_{2} − a_{3}) \\ = ∑_{n=max(a_{1},a_{2},a_{3})}^{min(b_{1},b_{2},b_{3})} \frac{(−1)^{n}}{(n − a_{1})!(n − a_{2})!(n − a_{3})!(b_{1} − n)!(b_{2} − n)!(b_{3} − n)!}, \end{matrix}  (87)

whose proof we omit. Inserting this into (84) gives the following formula, which is again symmetric in the indices “2” and “3” but in general has fewer terms than (84) or (86) (for instance, it reduces to a single term if any of the seven integers λ_{2}, λ_{3}, μ_{2}, μ_{3}, ν_{1}, ν_{2} or ν_{3} is 0):

Proposition 7. Let λ λ λ = (0, λ_{2}, λ_{3}) and μ μ μ = (μ_{1}, μ_{2}, μ_{3}) subject to the three inequalities ν_{i} ≥ 0, where ν_{1} = μ_{1} + λ_{2} + λ_{3}, ν_{2} = μ_{2} − λ_{2} + λ_{3}, ν_{3} = μ_{3} + λ_{2} − λ_{3}. Then

C(0, λ_{2}, λ_{3};μ μ μ) = (−1)^{λ_{3}} \frac{(μ_{2} + μ_{3} + s)! (μ_{1} + μ_{2} + λ_{3} + s)! (μ_{1} + μ_{3} + λ_{2} + s)!}{μ_{1}! (ν_{2} + s)! (ν_{3} + s)!} ×

∑_{n=max(0,λ_{2}−μ_{2},λ_{3}−μ_{3})}^{min(λ_{2},λ_{3})} \frac{(−1)^{n}}{(λ_{2} − n)!(λ_{3} − n)!(ν_{1} − n + s)!n!(μ_{2} − λ_{2} + n)!(μ_{3} − λ_{3} + n)!} .

Finally, we come to the general coefficient C(λ λ λ;μ μ μ). Equation (79) and induction on λ_{1} give

C(λ λ λ;μ μ μ) = \frac{(−1)^{λ_{1}} (μ_{2} + μ_{3} + λ_{1} + s)!}{μ_{2}! μ_{3}! (μ_{2} + μ_{3} + 2λ_{1} + s)!} ∑_{\begin{matrix}a,b,c≥0 \\ a+b+c=λ_{1} \end{matrix}} \frac{(μ_{2} + a + 2b)! (μ_{3} + a + 2c)!}{a! b! c!}

× C(0, λ_{2} − c, λ_{3} − b; μ_{1} − a, μ_{2} + a + 2b, μ_{3} + a + 2c) .

Substituting into this any of the three formulas obtained above for C(0, λ_{2}, λ_{3};μ μ μ) gives a formula for the general coefficient C(λ λ λ;μ μ μ) as a triple sum rather than an octuple one, which is what we would get from the generating function (69). For instance, the expression obtained using the formula in Proposition 7 is

C(λ λ λ;μ μ μ) = \frac{(μ_{2} + μ_{3} + λ_{1} + s)!}{μ_{2}!μ_{3}!(ν_{2} + s)!(ν_{3} + s)!}  ∑_{\begin{matrix}a,b,c≥0 \\ λ_{1}−μ_{1}≤b+c≤λ_{1} \\ λ_{2}−ν_{3}≤a+c≤λ_{2} \\ λ_{3}−ν_{2}≤a+b≤λ_{3} \end{matrix}} \frac{(−1)^{λ_{1}+a}}{a! b! c! (λ_{1} − b − c)! (λ_{2} − a − c)! (λ_{3} − a − b)!}

× \frac{(μ_{1} + μ_{2} + λ_{3} + b + s)!(μ_{1} + μ_{3} + λ_{2} + c + s)!(μ_{2} + λ_{1} + b − c)!(μ_{3} + λ_{1} − b + c)!}{(ν_{1} − a + s)! (μ_{1} − λ_{1} + b + c)! (ν_{2} − λ_{3} + a + b)! (ν_{3} − λ_{2} + a + c)!} .

This formula is symmetric in the indices “2” and “3”, but not in all three indices. Despite a fair amount of effort we were not able to simplify it or to find an expression for C(λ λ λ;μ μ μ) as a triple sum which is symmetric in all three indices. The formula for C(0, λ_{2}, λ_{3};μ μ μ) in Proposition 7 can be written as

C(0, λ_{2}, λ_{3};μ μ μ) = \frac{(μ_{2} + μ_{3} + s)! (μ_{1} + μ_{2} + λ_{3} + s)! (μ_{1} + μ_{3} + λ_{2} + s)!}{μ_{1}! μ_{2}! μ_{3}! (ν_{1} + s)! (ν_{2} + s)! (ν_{3} + s)!}
× Coeff_{Y^{λ_{2}} Z^{λ_{3}}} ((1 − Y)^{μ_{2}}(1 − Z)^{μ_{3}}(1 − Y Z)^{ν_{1}+s}) ,

which has a obvious generalization to

C(λ λ λ;μ μ μ) =^{?} ∏_{i=1}^{3} \frac{(λ_{i} + μ_{i+1} + μ_{i+1} + s)!}{μ_{i}! (ν_{i} + s)!} × Coeff_{X_{1}^{λ_{1}} X_{2}^{λ_{2}} X_{3}^{λ_{3}}} (∏_{i=1}^{3}(1 − X_{i})^{μ_{i}}(1 − X_{i+1}X_{i+2})^{ν_{i}+s}) .

but unfortunately this formula, which is symmetric and relatively simple, is not correct when all three λ_{i} are strictly positive; for instance, for C(1, 1, 1; 0, 0, 0) it gives 0 rather than the correct value of 4(s + 2).

We remark that the numbers S(a; b, c; d, e; f) are, up to simple normalizing factors, equal to the well-known Wigner 3j-symbols or Clebsch-Gordan coefficients occurring in the theory of angular momentum in quantum mechanics, and that an equality equivalent to (87) can be found in the physics literature. It is quite likely that the general coefficients C(λ λ λ;μ μ μ) can also be expressed in terms of the more complicated 6j-symbols (or Racah coefficients) and 9j-symbols of quantum mechanics, but we were not able to check whether this is the case.

E. Inhomogeneous coordinates and angular coordinates. As we already discussed at the end of §1, the homogeneity of the higher spherical polynomials makes it natural to consider them also as polynomials on the space S_{n}^{1} of n × n symmetric matrices with 1’s on the diagonal, or on the open subset S_{n}^{1,+} of this space consisting of positive definite matrices. For n = 3, this latter set can be identified by mapping t to the matrix T = T (t) defined in (7) with the semi-algebraic set

T = {t = (t_{1}, t_{2}, t_{3}) ∈ ℝ^{3} | |t_{i}| < 1, Δ(t) > 0}  (88)

shown in Figure 1 below, where Δ : ℝ^{3} → ℝ is the polynomial

Δ(t) = det T (t) = 1 − t_{1}^{2} − t_{2}^{2} − t_{3}^{2} + 2t_{1}t_{2}t_{3} .  (89)

(To see that t ↦ T (t) is surjective, recall that a symmetric matrix is positive definite if and only if its principal minors are positive. In fact only the top left minors are needed, so to define T it suffices to require Δ(t) > 0 and |t_{3}| < 1.) By Remark 4 after Theorem 2 in §3, we know that the higher spherical polynomials P_{ν ν ν}(T) = P_{ν ν ν,d}(T) for a fixed value of d = 2s + 4 > 2 then form an orthogonal Hilbert space basis of L^{2}(T , Δ^{s}dt_{1}dt_{2}dt_{3}). (Compare eq. 20). In this final subsection of §8 we wish to discuss a few further properties of the set T (alternative coordinates, symmetry, …). These properties will also play a role in [12], where we will study the differential equation (8) and its non-polynomial solutions.

We begin by mentioning two simple algebraic properties. The first is that the adjoint of the matrix T = T (t) (i.e., the matrix T^{∗} such that T T^{∗} = Δ · 1_{3}) is given by

T^{∗} = ( \begin{matrix}1 − t_{1}^{2} & Δ_{3} & Δ_{2} \\ Δ_{3} & 1 − t_{2}^{2} & Δ_{1} \\ Δ_{2} & Δ_{1} & 1 − t_{3}^{2} \end{matrix} ) , where Δ_{i} := \frac{1}{2} \frac{∂Δ}{∂t_{i}} = t_{j}t_{k} − t_{i} .  (90)

(Here and in the rest of this subsection, whenever we write indices i, j and k in the same formula we mean that {i, j, k} = {1, 2, 3}.) The other is that the determinant Δ has three algebraic factorizations

Δ = Δ_{1}^{+} Δ_{1}^{−} = Δ_{2}^{+} Δ_{2}^{−} = Δ_{3}^{+} Δ_{3}^{−} ,  (91)

where the quantities Δ_{i}^{±} are defined, using the convention and quantities just introduced, by

Δ_{i}^{±} = \sqrt{(1 − t_{j}^{2})(1 − t_{k}^{2})} ± Δ_{i} .  (92)

We will give an explanation and refinement of these factorizations below.

We now consider the symmetries of the set T , which involve a small surprise for which we have no real explanation. The polynomial Δ defined in (89) and the set T defined by (88) have a symmetry group G of order 24, given by the six permutations of the t_{i} together with the four changes of sign of the t_{i} preserving their product, i.e. G = 𝔖_{3} ⋉ E where

E = {εεε = (ε_{1}, ε_{2}, ε_{3}) ∈ {±1}^{3} | ε_{1}ε_{2}ε_{3} = 1 }

(Klein group). This group can be identified with 𝔖_{4} by identifying the set of indices {1, 2, 3} with the three possibile partitions of {1, 2, 3, 4} into two disjoint 2-tuples via i ↔ {{j, k}, {i, 4}} for {i, j, k} = {1, 2, 3}. The surprise is that the set T itself can be rewritten in a visibly

Figure 1: The semi-algebraic set T and the tetrahedron T_{θ}

𝔖_{4}-symmetric manner. We describe this here algebraically, and below in terms of angular coordinates. Let Ŝ_{4}^{1} denote the set of 4 × 4 real symmetric matrices that have 1’s on the diagonal and whose (i, j)- and (k, l)-entries sum to 0 whenever {i, j, k, l} = {1, 2, 3, 4}. The group 𝔖_{4} acts on Ŝ_{4}^{1} by simultaneous permutations of the rows and columns, and if we identify G with 𝔖_{4} as explained above, then we have an 𝔖_{4}-equivariant isomorphism S_{3}^{1} ≅ Ŝ_{4}^{1} given by

T = ( \begin{matrix}1 & t_{3} & t_{2} \\ t_{3} & 1 & t_{1} \\ t_{2} & t_{1} & 1 \end{matrix} ) ↔ T̂ = ( \begin{matrix}1 & t_{3} & t_{2} & −t_{1} \\ t_{3} & 1 & t_{1} & −t_{2} \\ t_{2} & t_{1} & 1 & −t_{3} \\ −t_{1} & −t_{2} & −t_{3} & 1 \end{matrix} ) .  (93)

The principal 2 × 2 minors of T̂ are just the numbers 1 − t_{i}^{2} and the principal 3 × 3 minors of T̂ are all equal to Δ, so the inequalities defining the set T say precisely that the principal 2 × 2 and 3 × 3 minors of T̂ are positive. This is an 𝔖_{4}-invariant condition and hence explains the 𝔖_{4}-symmetry of T (except that we cannot really explain the origin of the isomorphism (93)). Note that for the positive definiteness of the matrix T̂ (rather than just of its submatrix T) we need the positivity of all its principal minors, so that T̂ is positive definite if and only if T is and the determinant of T̂ is positive. This determinant, unlike that of T, has a rational factorization, as ∏_{ε ε ε∈E}(1 + εεε · t), the factors being the eigenvalues of the constant eigenvectors {(ε_{1}, ε_{2}, ε_{3}, −1)}_{ε ε ε∈E} of T̂. Finally, we mention that the adjoint of T̂ is a matrix of the same form as T̂ but with the diagonal entries 1 replaced by Δ and the off-diagonal entries t_{i} replaced by the expressions

t̂_{i} = t_{i} (1 + t_{j}^{2} + t_{k}^{2} − t_{i}^{2}) − 2t_{j}t_{k} = t_{i} Δ − 2Δ_{j}Δ_{k} .  (94)

These quantities will occur again in [12] in the construction of non-polynomal solutions of (8).

Our next topic, which again involves a surprise for which we have no real explanation, concerns angular coordinates. In the case of 2×2 symmetric matrices, corresponding to classical spherical polynomials, one frequently makes the change of variables t = cos θ to identify the space S_{2}^{1,+} = \{( \begin{matrix}1 & t \\ t & 1 \end{matrix} ) | −1 < t < 1\} with the interval (0, π). Here we make the corresponding substitution t_{i} = cos θ_{i} with θ_{i} ∈ (0, π). The surprise is that the semi-algebraic set T , which is shaped like a rounded tetrahedron, is then mapped isomorphically onto the linear tetrahedron

T_{θ} = {(θ_{1}, θ_{2}, θ_{3}) ∈ ℝ^{3} | θ_{i} < θ_{j} + θ_{k} < 2π − θ_{i} for {i, j, k} = {1, 2, 3} } ,  (95)

which is the interior of the convex hull of the four vertices (0, 0, 0), (0, π, π), (π, 0, π) and (π, π, 0) of the cube [0, π]^{3}. (See Figure 1.) This gives another explanation of the isomorphism between the symmetry group of T and 𝔖_{4}, identified with the group of permutations of these four vertices. But we can do even better: if we introduce new coordinates δ_{ν} (ν = 1, 2, 3, 4) by

δ_{i} = \frac{θ_{j} + θ_{k} − θ_{i}}{2} (1 ≤ i ≤ 3),  δ_{4} = π − \frac{θ_{1} + θ_{2} + θ_{3}}{2} ,  (96)

then the angles θ_{i} can be expressed in terms of the δ_{ν} by

θ_{i} = δ_{j} + δ_{k} ,  π − θ_{i} = δ_{i} + δ_{4}  (1 ≤ i ≤ 3),  (97)

the tetrahedron T_{θ} is identified with the simplex {(δ_{1}, δ_{2}, δ_{3}, δ_{4}) ∈ ℝ_{>0}^{4} | δ_{1} + δ_{2} + δ_{3} + δ_{4} = π}, and G acts simply by permuting the δ’s. The six numbers {±t_{i}}_{1≤i≤3} are the six numbers {cos(δ_{μ} + δ_{ν})}_{1≤μ<ν≤4}, these being precisely the non-diagonal entries of the matrix T̂ defined in (93) and with the same numbering. Finally, the three factorizations of Δ given in (91) can be replaced by the single fourfold factorization

Δ = 4 ∏_{ν=1}^{4} sin δ_{ν} ,  (98)

which explains and refines (91) because the functions Δ_{i}^{−} and Δ_{i}^{+} are equal to 2 sin δ_{i} sin δ_{4} and 2 sin δ_{j} sin δ_{k}, respectively. This equation also makes it clear that the condition Δ > 0 is equivalent to the inequalities δ_{ν} > 0 and hence explains why the sets T and T_{θ} correspond.

§9. A universal generating function for the descending basis

In Remark 1 in §8.A we observed that the generating function of Theorem 9 could be expressed in terms of only three quantities, namely the coefficients of the characteristic polynomial of the product of T with a 3×3 matrix of “dummy” variables X_{j} . That remark will now be generalized to construct a generating function for all n whose coefficients give a basis of the space P^{(n)}(d) for all n and d, and in fact a basis that coincides, up to scalar factors, with the “descending” basis of P^{(n)}(d) defined in Chapter I.

We fix n (at least initially), but take d to be generic. Equivalently, we work over the field K = ℚ(d) rather than thinking of d as a specific complex number. (Compare the discussion following Theorem 6 in §5.) We would like to construct a generating function over K[S_{n}] whose coefficients P_{ν ν ν}(T) are multiples of the descending basis, i.e., which satisfy

D_{ij} P_{ν ν ν}(T) =^{.} P_{ν ν ν−e_{ij}}(T)  (99)

for all ννν ∈ N_{0} and all i, j = 1, …, n. Here =^{.} denotes equality up to a non-zero constant and we are also using the convention that P_{ν ν ν}=0 if some entry of ννν is negative, so (99) includes the fact that each P_{ν ν ν} belongs to P^{(n)}(d). Generalizing the observation about Theorem 9 quoted above, we make the following Ansatz for the form of this generating function:

∑_{ν ν ν∈N_{0}} P_{ν ν ν}(T) X^{ν ν ν} = G^{(n)}(σ_{1}(XT), σ_{2}(XT), …, σ_{n}(XT)) ,  (100)

where G^{(n)} is some power series in n variables, σ_{a} of a square matrix denotes the ath elementary symmetric function of its eigenvalues, X is a symmetric n × n matrix of “dummy” variables x_{ij} = x_{ji} with x_{ii} = 0 for all i, and X^{ν ν ν} = ∏_{i≠j} x_{ij}^{ν_{ij}/2} = ∏_{i<j} x_{ij}^{ν_{ij}}. Our object is to show that a function G^{(n)}(σ_{1}, …, σ_{n}) can be found such that the coefficients P_{ν ν ν} defined by (100) satisfy the property (99). By the Remark cited above, we already know the answer for n = 3, namely

G^{(3)}(σ_{1}, σ_{2}, σ_{3}) = \frac{1}{\sqrt{Δ_{0}^{2} − 8σ_{3}}} (\frac{Δ_{0} + \sqrt{Δ_{0}^{2} − 8σ_{3}}}{2} )^{− \frac{d−4}{2}} , Δ_{0} := (1 − \frac{σ_{1}}{2} )^{2} − σ_{2} . (101)

It will turn out that the functions G^{(n)} are actually independent of n in the sense that there is a single power series G(σ_{1}, σ_{2}, …) in infinitely many variables such that each G^{(n)}(σ_{1}, …, σ_{n}) is simply G(σ_{1}, …, σ_{n}, 0, 0, …). (Notice that such a power series makes sense because we can introduce a grading by |σ_{a}| = a and then the space of monomials of any fixed degree is finite-dimensional.) The power series G turns out not to be quite unique, but to be uniquely determined by G^{(1)}(σ_{1}) = G(σ_{1}, 0, 0, …), which is arbitrary. The choice corresponding to (101) in the case n = 3 and to the classical generating function of Gegenbauer polynomials in the case n = 2 is G^{(1)}(σ_{1}) = (1 − σ_{1}/2)^{2−d}, and this will be our standard choice, but there exist at least two other special choices which have some nice properties, as will be discussed briefly in Subsection C.

Since this section is again quite long, we have divided it into subsections. In Subsection A we show how to characterize the power series G^{(n)} such that the Taylor coefficients P_{ν ν ν} in (100) satisfy (99), and prove that for each k the space of the degree k parts of such power series is one-dimensional, which implies the desired existence and uniqueness results. The proofs of two of the propositions needed for this are quite long and are given separately in Subsection B, which can be omitted without loss of continuity. One consequence of the proof is that if G^{(1)} (which is arbitrary) is chosen to be holomorphic in d, then the coefficients of the power series G^{(n)} have poles only at integral values of d. This implies the result, left open in §5, that the scalar product on P^{(n)}(d) is non-degenerate for all d ∈ ℂ ∖ ℤ_{<n}. Subsection A also contains an explicit inductive construction of the power series G^{(n)} from an arbitrary initial value of G^{(1)} by applying suitable differential operators. Finally, in Subsection C we give a discussion of the various good choices of G^{(1)} and a number of examples and partial results about the coefficients of the power series G^{(n)} for the standard choice. We have not been able to find a complete formula for the coefficients in general.

A. Existence of the power series G^{(n)}. We fix the notation

V = K[[σ_{1}, σ_{2}, σ_{3}, …]], V_{k} = V_{deg=k} ,

where K = ℚ(d) as before and the degrees are determined by |σ_{a}| = a. Thus V_{k} is finitedimensional, of dimension p(k) (number of partitions of k), and V ≅ ∏_{k≥0} V_{k}. We have a homomorphism V → K[S_{n}][[X]] defined by F ↦ F̃, where

F̃(T, X) := F (σ_{1}(XT), σ_{2}(XT), …, σ_{n}(XT), 0, 0, …) .

Our goal is to find a G^{(n)} such that the coefficients P_{ν ν ν} in (100) satisfy (99), so the first thing we need is to compute the action of all D_{ij} on F̃ for arbitrary F. (Note that the operators D_{ij} do not involve the X-variables, so D_{ij}(G^{(n)}) = ∑(D_{ij}P_{ν ν ν})X^{ν ν ν}.) The image of the map F ↦ F̃ is not stable under D_{ij}, but the following proposition, whose proof will be given in Subsection B, calculates D_{ij}(F̃) for all F ∈ V in terms of only the quantities ∂_{ij}(σ_{a}) and certain differential operators L_{p} that do preserve V. For convenience we set ∂_{a} = ∂/∂σ_{a}, while ∂_{ij} retains its usual meaning (1 + δ_{ij}) ∂/∂t_{ij} .

Proposition 1. For p ≥ 1, define a second order differential operator L_{p} : V → V of degree −p (i.e., L_{p}(V_{k}) ⊆ V_{k−p}) by

L_{p} = (d + 1 − p) ∂_{p} + 2 ( ∑_{a, b≥p} − ∑_{0<a, b<p} ) σ_{a+b−p} ∂_{a} ∂_{b} ,  (102)

where σ_{q} is taken to be 1 if q = 0 and 0 if q < 0. Then for all i and j we have

D_{ij} = ∑_{p=1}^{∞} ∂_{ij}(σ_{p}) L_{p} .  (103)

The more precise meaning of (103) for a given value of n is that

D_{ij}(F̃) = ∑_{p=1}^{n} ∂_{ij}(σ_{p}(XT)) { \atop L_{p}(F)} (1 ≤ i, j ≤ n) ,

where the sum over p can be extended to infinity if one likes since σ_{p}(XT) = 0 for p > n. Note that here one could replace the sums in (102) by the subsums with 1 ≤ a, b ≤ n and a + b − p ≤ n, since if a + b − p > n then σ_{a+b−p} vanishes, and if a + b − p ≤ n then a and b are ≤ n automatically in the first sum and also in the second sum whenever p ≤ n. Similar remarks apply to all later formulas, which we will write in a way not explicitly mentioning n, but remembering that, when we apply them to σ_{i} = σ_{i}(XT) for matrices X and T of a fixed size n, then σ_{a} = 0 for all a > n.

Corollary. Let W = ⋂_{p≥2} Ker(L_{p}) ⊂ V. Then for any F ∈ W we have D_{ii}(F̃) = 0 and

D_{ij}(F̃) = 2x_{ij} { \atop L_{1}(F)} for i ≠ j.

Proof. This follows immediately from (103), since ∂_{ii}(σ_{1}(XT)) = 0 and ∂_{ij}(σ_{1}(XT)) = 2 x_{ij}. □

The following proposition, whose proof will also be postponed to Subsection B, is crucial.

Proposition 2. The operators L_{p} satisfy the commutation formula

[L_{p_{1}}, L_{p_{2}}] = −2 ∑_{p_{1}<p≤p_{2}} ∂_{p_{1}+p_{2}−p} L_{p}  (0 < p_{1} < p_{2}) .  (104)

Corollary. The subspace W of V is mapped into itself by L_{1}. More generally, the space W ⟨m⟩ := ⋂_{p>m} Ker(L_{p}) ⊂ V is mapped into itself by L_{m} for all m ≥ 1.

Proof. This follows immediately from (104). In fact, we need only that the product L_{p_{2}}L_{p_{1}} is in the left ideal generated by L_{p}’s with p > p_{1}. □

We now use this to deduce the main result of this subsection:

Theorem 10. The space W_{k} = W ∩ V_{k} is one-dimensional for each k ≥ 0. If G_{k} is a non-zero element of W_{k} for every k ≥ 0, then L_{1}(G_{k}) is a non-zero multiple of G_{k−1} for all k ≥ 1 and if we set G = ∑_{k} G_{k} ∈ W then the polynomials P_{ν ν ν} ∈ K[S_{n}] defined by G = ∑ P_{ν ν ν}(T)X^{ν ν ν} satisfy (99).

Proof. For each m ≥ 1 we denote by V^{(m)} the ring K[σ_{1}, …, σ_{m}] and by ρ_{m} : V → V^{(m)} the restriction map G ↦ G(σ_{1}, …, σ_{m}, 0, 0, …). We will show by downwards induction on m that for each m ≥ 0 the composite map

ρ_{m}|_{W ⟨m⟩} : W ⟨m⟩ ⟶ V ⟶^{ρ_{m}} V^{(m)}  (105)

is an isomorphism. The case m = 1 gives the first assertion of the theorem, since V^{(1)} = K[σ_{1}] is 1-dimensional over K in each degree. Here “downward induction” makes sense because both maps (105) preserves degree and are obviously isomorphisms in degree k when m > k, so that the “initial step” of the induction is automatically satisfied.

We will show below that the map (105) is injective. For the surjectivity we use a dimension argument. Denote by W_{k}⟨m⟩ and V_{k}^{(m)} the degree k parts of W ⟨m⟩ and V^{(m)}, respectively. The dimension of V_{k}^{(m)} is the number p_{m}(k) of partitions of k into parts ≤ m. These numbers are given by the generating function

∑_{k=0}^{∞} p_{m}(k) t^{k} = \frac{1}{(1 − t)(1 − t^{2}) ⋯ (1 − t^{m})} ,

and from this or directly from the definition one sees that they satisfy the recursion

p_{m}(k) = p_{m}(k − m) + p_{m−1}(k)  (106)

(with the initial condition p_{0}(k) = δ_{k0} and the convention p_{m}(k) = 0 for k < 0). On the other hand, L_{m} acts on W ⟨m⟩ by the corollary to Proposition 2, clearly with kernel W ⟨m − 1⟩, so since L_{m} has degree −m we have an exact sequence

0 ⟶ W_{k}⟨m − 1⟩ ⟶ W_{k}⟨m⟩ ⟶^{L_{m}} W_{k−m}⟨m⟩  (107)

for every m and k (with the convention that the last space is {0} if m > k). If we assume that the map (105) is an isomorphism for m, then it follows from (107) and (106) that dim W_{k}⟨m − 1⟩ ≥ p_{m−1}(k) = dim V_{k}^{(m−1)}, and hence, assuming that the injectivity is known in general, that the map (105) with m replaced by m − 1 is an isomorphism, completing the induction.

It remains to prove the injectivity in (105). To do this we will show that

Ker(L_{p}) ∩ Ker(ρ_{p−1}) ⊆ Ker(ρ_{p})  (108)

for all p. Then for F ∈ W ⟨m⟩ = Ker(L_{m+1}) ∩ Ker(L_{m+2}) ∩ ⋯ we can apply (108) successively with p = m + 1, m + 2, … to get ρ_{m}(F) = 0 ⇒ ρ_{m+1}(F) = 0 ⇒ ρ_{m+2}(F) = 0 ⇒ ⋯ and hence F =0, proving the injectivity of ρ_{m}.

To prove (108), write the expansion of an arbitrary element F ∈ V as

F (σ_{1}, σ_{2}, …) = ∑_{r≥0} A(r) \frac{σ^{r}}{r!!!} = ∑_{r_{1},r_{2},···≥0} A(r_{1}, r_{2}, …) \frac{σ_{1}^{r_{1}}}{r_{1}!} \frac{σ_{2}^{r_{2}}}{r_{2}!} …,  (109)

where r runs over ∞-tuples (r_{1}, r_{2}, …) with r_{i} ≥ 0 for all i and r_{i} = 0 for all but finitely many i. (The coefficient A(r) is an element of K ⊂ ℚ(d) and will be denoted A(r; d) when we want to make the dependence on d explicit.) We also make the convention

r_{0} = 1 ,  r_{−1} = r_{−2} = ⋯ = 0 ,  (110)

and will also write simply A(r_{1}, …, r_{p}) for A(r_{1}, …, r_{p}, 0, 0, …) if r_{i} = 0 for all i > p. Studying ρ_{p}(F) means looking only at the coeficients A(r_{1}, …, r_{p}) in the expansion (109). If F ∈ Ker(L_{p}),

then by computing the coefficient of \frac{σ_{1}^{r_{1}}}{r_{1}!} … ^{\frac{σ_{p−1}^{r_{p−1}}}{r_{p−1}!}} \frac{σ_{p}^{rp−1}}{(r_{p}−1)!} in L_{p}(F) we find

A(r) = \frac{2}{d − p + 2r_{p} − 1} ∑_{\begin{matrix}0<a, b<p \\ a+b≥p \end{matrix}} r_{a+b−p} A(r − e_{a+b−p} + e_{a} + e_{b} − e_{p})  (111)

for all r = (r_{1}, …, r_{p}) with r_{p} > 0, where e_{i} as usual denotes the vector with a 1 in the ith place and 0’s elsewhere (so e_{0} = 0) and r_{0} = 1 by the convention (110). From this formula it follows inductively that all coefficients A(r_{1}, …, r_{p}) are determined by the coefficients A(r_{1}, …, r_{p−1}) and hence that ρ_{p−1}(F) = 0 ⇒ ρ_{p}(F) = 0, as was to be proved.

This completes the proof of the isomorphy of the map (105) and hence of the first statement of Theorem 10. The other two statements are now easy. Choose a basis {G_{k}} of the 1-dimensional space W_{k} for each k ≥ 0. From the corollary to Proposition 2 we see that L_{1}(G_{k}) is a multiple of G_{k−1} for all k ≥ 1, and this multiple is non-zero because of (107), since W_{k}⟨0⟩ ≅ V_{k}^{(0)} = {0} for k > 0. Write L_{1}(G_{k}) = c_{k}G_{k−1} and write G = ∑_{k=0}^{∞} G_{k} as ∑_{ν ν ν} P_{ν ν ν}(T)X^{ν ν ν} . By the corollary to Proposition 1 we have D_{i}(G̃_{k}) = 0 and D_{ij}(G̃_{k}) = 2c_{k}x_{ij}G̃_{k−1} , and comparing the coefficients of X^{ν ν ν} on both sides of this equality we find that D_{i}(P_{ν ν ν}) = 0 and D_{ij}(P_{ν ν ν}) = 2c_{∥ν ν ν∥}P_{ν ν ν−e_{ij}} for all ννν ≠ 0, where ∥ννν∥ = ∑_{i<j} ν_{ij} . □

Before finishing this subsection, we mention two consequences of the explicit recursion (111). First, in this formula, the only denominators that occur are of the form d − d_{0} with d_{0} ∈ ℤ. It follows that, if we choose our normalization of the functions G_{k} ∈ W_{k} (or equivalently, of the function G^{(1)}) to have coefficients in the subring

R = ℚ[d, \frac{1}{d}, \frac{1}{d ± 1}, \frac{1}{d ± 2}, …]  (112)

of K consisting of rational functions of d having poles only at integers, then the whole function G has coefficients in this ring. The coefficients c_{k} ∈ K relating L_{1}(G_{k}) and G_{k−1} also have no zeros or poles in d outside of ℤ, and it follows that the descending basis elements P_{ν ν ν}^{D}(T) that we constructed in §5 also have coefficients in the ring R. In particular, these polynomials exist for any d ∈ ℂ ∖ ℤ. It then follows from the duality between the bases {P_{ν ν ν}^{D}} and {P_{ν ν ν}^{M}} (Theorem 6 and the remarks immediately preceding and following it) that the scalar product in P^{(n)}(d) is non-degenerate for these values of d. But we already know that the scalar product is non-degenerate for real d > n − 1, because it is defined by a convergent integral with strictly positive integrand (cf. Theorem 2), so we deduce the following result, already mentioned in §5:

Theorem 11. The scalar product in P^{(n)}(d) is non-degenerate for all d ∈ ℂ ∖ ℤ_{<n}.

We remark that the non-degeneracy for d ∈ ℤ_{≥n} can also be obtained purely algebraically, without using the positivity of the scalar product, since the denominator d − d_{0} in (111) always satisfies d_{0} < p (because r_{p} ≥ 1) and since p ≤ n for the coefficients A(r) occurring in G^{(n)}.

The second observation is that the recursion (111) leads relatively easily to a “closed formula” for the generating series G(σ_{1}, σ_{2}, …). Indeed, for each n, let us expand the generating function G^{(n)} = ρ_{n}(G) (n > 1) as a power series

G^{(n)}(σ_{1}, …, σ_{n}) = ∑_{r=0}^{∞} g_{r}^{(n)}(σ_{1}, …, σ_{n−1}) \frac{σ_{n}^{r}}{r!} (g_{0}^{(n)} = G^{(n−1)})

in its last variable σ_{n}. Applied to any function involving only σ_{1}, …, σ_{n}, the operator L_{n} equals (d + 1 − n)∂_{n} + 2σ_{n}∂_{n}^{2} − 2M_{n}, where M_{n} is the second order differential operator

M_{n} = ∑_{\begin{matrix}0<a, b<n \\ a+b≥n \end{matrix}} σ_{a+b−n} ∂_{a} ∂_{b} ,  (113)

with σ_{0} = 1 as usual. Equating the coefficient of σ_{n}^{r} in L_{n}(G^{(n)}) to 0 therefore gives

\frac{d − n + 2r + 1}{2} g_{r+1}^{(n)} = M_{n}(g_{r}^{(n)})  (r ≥ 0)  (114)

and hence, by induction on r and using the initial condition g_{0}^{(n)} = G^{(n−1)}, the general formula

g_{r}^{(n)} = \frac{1}{(\frac{1}{2}(d − n + 1))_{r}} M_{n}^{r}(G^{(n−1)}) (r ≥ 0) .

Since the operators M_{n} and (multiplication by) σ_{n} commute, this implies that

G^{(n)}(σ_{1}, …, σ_{n}) = 𝕁_{\frac{d−n−1}{2}} (σ_{n}M_{n})(G^{(n−1)}(σ_{1}, …, σ_{n−1})) ,  (115)

where 𝕁_{ν}(x) is the power series

𝕁_{ν}(x) = ∑_{r=0}^{∞} \frac{x^{r}}{r! (ν + 1)_{r}} = 1 + \frac{x}{ν + 1} + \frac{x^{2}}{2(ν + 1)(ν + 2)} + ⋯,  (116)

related to the J-Bessel function J_{ν}(x) by J_{ν}(x) = \frac{(x/2)^{ν}}{Γ(ν+1)}𝕁_{ν}(−x^{2}/4). Now induction on n gives

G^{(n)}(σ_{1}, …, σ_{n}) = 𝕁_{\frac{d−n−1}{2}} (σ_{n}M_{n}) 𝕁_{\frac{d−n}{2}} (σ_{n−1}M_{n−1}) ⋯ 𝕁_{\frac{d−3}{2}} (σ_{2}M_{2})(G^{(1)}(σ_{1})) , (117)

the desired “closed formula” for G^{(n)} (or even, if we extend the product of operators infinitely to the left, for the whole generating function G). The form of (117) makes it clear that the power series G^{(1)}(σ_{1}) can be chosen arbitrarily but then determines all of the higher power series G^{(n)}(σ_{1}, …, σ_{n}). Summarizing, we have:

Theorem 12. The universal generating function G^{(n)} can be obtained for all n ≥ 1 from an arbitrary choice of G^{(1)} by (117), where 𝕁_{ν}(x) and M_{n} are the power series and second order differential operator defined by (116) and (113), respectively.

The above argument described the generating function G^{(n)} for generic values of d in terms of the initial choice G^{(1)}, the key point being that G^{(n)} is determined in terms of G^{(n−1)} by equation (115) as long as (d−n−1)/2 is not a negative integer. We end this subsection by saying something about the exceptional cases where G(σ_{1}, …, σ_{n}) is not determined by its specialization to σ_{n} = 0. For clarity we write W^{(n)}(d) for the space of functions in V^{(n)} = K[[σ_{1}, …, σ_{n}]] annihilated by L_{2}, …, L_{n}, including the dependence on d in the notation. Then we have:

Proposition 3. Suppose that G(σ_{1}, …, σ_{n}) ∈ W^{(n)}(d)∖{0} with G(σ_{1}, …, σ_{n−1}, 0) ≡ 0. Then d = n + 1 − 2m for some integer m ≥ 1 and G has the form

G(σ_{1}, …, σ_{n}) = ∑_{ν=0}^{∞} M_{n}^{ν}(g)(σ_{1}, …, σ_{n−1}) \frac{σ_{n}^{m+ν}}{ν! (m + ν)!}  (118)

for some function g ∈ W^{(n−1)}(d + 4m) = W^{(n−1)}(n + 2m + 1). Conversely, for any m ∈ ℤ_{≥1} and any g ∈ W^{(n−1)}(n + 2m + 1), the function G defined by (118) belongs to W^{(n)}(n − 2m + 1).

Proof. Write G as ∑_{r≥0} g_{r}(σ_{1}, …, σ_{n−1}) σ_{n}^{r}/r!, and let m ≥ 1 be the smallest integer for which

g_{m} ≢ 0. From L_{n}^{(n,d)}(G) = 0 we get \frac{1}{2}(d−n+2r+1)g_{r+1} = M_{n}(g_{r}) for all r ≥ 0 (equation (114)). Applying this with r = m−1 gives d = n−2m+1, and then applying it inductively for r = m+ν gives g_{m+ν} = M_{n}^{ν}(g_{m})/ν! for all ν ≥ 0, so G has an expansion as in (118) with g = g_{m}. To prove that the function g belongs to W^{(n−1)}(d + 4m), we must show that L_{p}^{(n−1,d+4m)}g = 0 for all 2 ≤ p < n, where for clarity we write L_{p}^{(n,d)} for the operator L_{p} in V^{(n)} with a given value of d. From the definition (102) we get

L_{p}^{(n,d)} = L_{p}^{(n−1,d+4m)} + 4(σ_{n} \frac{∂}{∂σ_{n}} − m) \frac{∂}{∂σ_{p}} + 2σ_{n} ∑_{p<i<n} \frac{∂^{2}}{∂σ_{i} ∂σ_{n+p−i}} (2 ≤ p < n),

and the claim then follows by computing the coefficient of σ_{n}^{m} in the equation L_{p}^{(n,d)}(G) = 0. Conversely, if g ∈ W^{(n−1)}(n + 2m + 1), then by reversing the above calculations we find that the function G defined by (118) satisfies G = O(σ_{n}^{m}), G ∈ Ker(L_{n}^{(n,d)}), and L_{p}^{(n,d)}(G) = O(σ_{n}^{m+1}) for 2 ≤ p < n. But by the corollary to Proposition 2, L_{p}^{(n,d)}(G) for 2 ≤ p < n also belongs to Ker(L_{n}^{(n,d)}), and then it must vanish since the first part of the proof shows that any element of this kernel for d = n + 1 − 2m is determined by its σ_{n}^{m} term. □

B. Proofs of the closed formula and commutation relation for L_{p}. In this subsection we prove Propositions 1 and 2. Both proofs are rather lengthy and, as already mentioned, can be omitted without loss of continuity. To simplify them somewhat, we define coefficients ε(m), ε_{a,b}^{±}(m) ∈ {0, ±1, ±2} for a, b, m ∈ ℤ as follows:

ε(m) = sign(m + \frac{1}{2}) = \{ \begin{matrix}+1 & if m ≥ 0, \\ −1 & if m < 0, \end{matrix}

ε_{a,b}^{+}(m) = ε(a − m) + ε(b − m) = \{ \begin{matrix}+2 & if a, b ≥ m, \\ −2 & if a, b < m, \\ 0 & otherwise, \end{matrix}

ε_{a,b}^{−}(m) = ε(a − m) − ε(b − m) = \{ \begin{matrix}+2 & if b < m ≤ a, \\ −2 & if a < m ≤ b, \\ 0 & otherwise. \end{matrix}

Then the definition (102) of the operators L_{p} on V can be written more compactly as

L_{p} = (d + 1 − p) ∂_{p} + ∑_{a,b≥1} ε_{a,b}^{+}(p) σ_{a+b−p} ∂_{a} ∂_{b}  (p ≥ 1)  (119)

(with ∂_{a} = ∂/∂σ_{a}, and with the convention that σ_{0} = 1 and σ_{a} = 0 if a < 0), and the commutation relation (104) can be written as

[L_{p}, L_{q}] = ∑_{\begin{matrix}a, b≥1 \\ a+b=p+q \end{matrix}} ε_{p,q}^{−}(b) ∂_{a} L_{b}  (p, q ≥ 1).  (120)

Proof of Proposition 1. Since both the left- and right-hand sides of (103) are second order differential operators, it suffices to prove their equality on monomials in K[σ_{1}, σ_{2}, …] of degree ≤ 2, i.e., for 1, σ_{a} and σ_{a}σ_{b} (a, b ≥ 1). The first case is trivial (both sides are zero), the second case can be rewritten (since L_{p}(σ_{a}) = (d + 1 − p)δ_{ap}) in the form

D_{ij}(σ_{p}) = (d + 1 − p) ∂_{ij}(σ_{p})  (1 ≤ i, j ≤ n, 1 ≤ p ≤ n) ,  (121)

and the third case is equivalent, by virtue of (119) and (121), to the identity

D_{ij}(σ_{a}σ_{b}) = ∑_{p=1}^{∞} ∂_{ij}(σ_{p}) [(d + 1 − p)(δ_{ap}σ_{b} + δ_{bp}σ_{a}) + 2 ε_{a,b}^{+}(p)σ_{a+b−p}]
= D_{ij}(σ_{a}) σ_{b} + σ_{a} D_{ij}(b) + 2 ∑_{p≥1} ε_{a,b}^{+}(p) ∂_{ij}(σ_{p}) σ_{a+b−p} ,  (122)

It therefore suffices to prove the formulas (121) and (122).

For this purpose, it will be convenient to use generating functions. Let λ be an independent variable and set

P = P_{λ} = X^{−1} + λT , Δ = Δ_{λ} = det(P_{λ}) = det(X)^{−1} ∑_{p=0}^{n} σ_{p} λ^{p} .

The subscript “λ” will be omitted when no confusion can result. Since the factor det(X)^{−1} is independent of λ and the T -variables, (121) can be rewritten in the form

D_{ij}^{0}(Δ) = (1 − λ \frac{d}{dλ}) ∂_{ij}(Δ) ,  (123)

where D_{ij}^{0} := D_{ij} − d∂_{ij} denotes the purely second degree part of the differential operator D_{ij}. (Cf. the definition of D_{ij} in eq. (31).) Since the (i, j)-entry P_{ij} of P equals λt_{ij} + (X^{−1})_{ij}, we see from the expansion of det(P) as a sum of products that

∂_{ij}(Δ) = 2 λ Δ^{i;j} ,  (124)

where Δ^{i;j} is defined as (−1)^{i+j} times the determinant of the (n − 1) × (n − 1) matrix P^{i;j} obtained from P by omitting its ith row and jth column. (The factor 2 in (124) arises for i ≠ j because P is symmetric and contains t_{ij} twice, and for i = j because ∂_{ii} = 2 ∂/∂t_{ii}.) Since each entry of the matrix P^{i;j} = λT^{i;j} + (X^{−1})^{i;j} contains λ multiplied by the corresponding entry of T^{i;j}, which is some t_{kℓ} with k ≠ i, ℓ ≠ j, we have

\frac{∂}{∂λ}(Δ^{i;j}) = ∑_{k, ℓ=1}^{n} t_{kℓ} Δ^{i,k;j,ℓ} ,  (125)

where Δ^{i,k;j,ℓ} is the determinant of the (n − 2) × (n − 2) matrix obtained from P by omitting its ith and kth rows and jth and ℓth columns, multiplied by (−1)^{i+k+j+ℓ} and by an additional factor sgn(i − k) sgn(j − ℓ) to take into account the antisymmetry of the determinant (and Δ^{i,k;j,ℓ} = 0 if i = k or j = ℓ). Combining (124) and (125) gives

(1 − λ \frac{d}{dλ}) ∂_{ij}(Δ) = −2 λ^{2} \frac{∂}{∂λ}(Δ^{i;j}) = −2 λ^{2} ∑_{k, ℓ} t_{kℓ} Δ^{i,k;j,ℓ} .  (126)

On the other hand, using (124) with (i, j) replaced by (j, ℓ) and computing ∂_{ik} of it by the same argument, we find

∂_{ik}∂_{jℓ}(Δ) = 2 λ ∂_{ik}(Δ^{j;ℓ}) = 2λ^{2} [Δ^{i,j;k,ℓ} + Δ^{k,j;i,ℓ}] .  (127)

(Here there are two terms on the right-hand side rather than a factor of 2 as in (124), because P^{j;ℓ} is no longer symmetric.) Multiplying this by t_{kℓ} and summing over all 1 ≤ k, ℓ, ≤ n gives

D_{ij}^{0}(Δ) = ∑_{k, ℓ} t_{kℓ} ∂_{ik}∂_{jℓ}(Δ) = 2λ^{2} ∑_{k,ℓ} t_{kℓ} Δ^{k,j;i,ℓ} .  (128)

(The whole contribution from the first term on the right-hand side of (127) drops out because it is antisymmetric in k and ℓ while t_{kℓ} is symmetric.) Interchanging i and j in this formula and using the antisymmetry once again, we see that the right-hand sides of (126) and (128) agree, completing the proof of (123) and hence of (121).

We now turn to equation (122). Again we use the generating function Δ_{λ} = det(P_{λ}), but this time for two independent variables λ and μ (which we now cannot omit from the notations). From the definition of D_{ij} together with equation (124), which by the standard formula for the inverse of a matrix can be written in the form ∂_{ij}(Δ_{λ}) = 2λΔ_{λ}(P_{λ}^{−1})_{ij}, we find

D_{ij}(Δ_{λ}Δ_{μ}) − D_{ij}(Δ_{λ}) Δ_{μ} − Δ_{μ} D_{ij}(Δ_{l}) = ∑_{k, ℓ} t_{kℓ} (∂_{ik}(Δ_{l})∂_{jℓ}(Δ_{μ}) + ∂_{ik}(Δ_{μ})∂_{jℓ}(Δ_{λ}))

= 4 λ μ Δ_{λ} Δ_{μ} ∑_{k, ℓ}((P_{λ}^{−1})_{ik} t_{kℓ} (P_{μ}^{−1})_{ℓj} + (P_{μ}^{−1})_{ik} t_{kℓ} (P_{λ}^{−1})_{ℓj})

= 4 λ μ Δ_{λ} Δ_{μ} (P_{λ}^{−1} T P_{μ}^{−1} + P_{μ}^{−1} T P_{λ}^{−1})_{ij}

= −8 λ μ Δ_{λ} Δ_{μ} (\frac{P_{λ}^{−1} − P_{μ}^{−1}}{λ − μ} )_{ij} (because T = \frac{P_{λ} − P_{μ}}{λ − μ} )

= −\frac{4}{λ − μ} (μ Δ_{μ} ∂_{ij}(Δ_{λ}) − λ Δ_{λ} ∂_{ij}(Δ_{μ}))

= \frac{4}{det(X)^{2}} ∑_{p≥0, q≥0} σ_{q} ∂_{ij}(σ_{p}) \frac{λ^{q+1}μ^{p} − μ^{q+1}λ^{p}}{λ − μ} ,

and now expanding the fraction in the last line as a geometric series and comparing the coefficients of λ^{a}μ^{b} on both sides we obtain the desired equation (122). □

Proof of Proposition 2. Any element of the Weyl algebra ℚ[σ_{a}, σ_{2}, …, ∂_{1}, ∂_{2}, …] can be written uniquely using the commutation relation [∂_{a}, σ_{b}] = δ_{ab} as a sum of monomials σ_{1}^{∗}σ_{2}^{∗} ⋯ ∂_{1}^{∗}∂_{2}^{∗} ⋯, so it suffices to put the left- and right-hand sides of equation (120) into this form and compare coefficients. We claim that both sides of (120) are equal to

2 ∑_{\begin{matrix}i, j≥1 \\ i+j=p+q \end{matrix}}(d + 1 − i) ε_{p,q}^{−}(i) ∂_{i} ∂_{j} + ∑_{i, j, k≥1} ε_{p,q}^{−}(p + q − k) ε_{i,j}^{+}(p + q − k) σ_{i+j+k−p−q} ∂_{i}∂_{j}∂_{k} .

For the left side we use the commutation relation to find in succession

[∂_{p}, L_{q}] = ∑_{a+b=p+q} ε_{a,b}^{+}(q) ∂_{a}∂_{b} ,

[σ_{i}∂_{a}∂_{b}, σ_{I}∂_{A}∂_{B}] = σ_{i} (δ_{b,I}∂_{a} + δ_{a,I}∂_{b})∂_{A}∂_{B} − σ_{I} (δ_{B,i}∂_{A} + δ_{A,i}∂_{B})∂_{a}∂_{b} ,

[L_{p}, L_{q}] = ∑_{i+j=p+q}[(d + 1 − p)ε_{i,j}^{+}(q) − (d + 1 − q)ε_{i,j}^{+}(p)] ∂_{i}∂_{j}

+ ∑_{i, j, k>0}(ε_{i,j}^{+}(q) ε_{k,i+j−q}^{+}(p) − ε_{i,j}^{+}(p) ε_{k,i+j−p}^{+}(q)) σ_{i+j+k−p−q} ∂_{i}∂_{j}∂_{k} .

The final expression gives [L_{p}, L_{q}] as a sum of canonically ordered monomials in the Weyl algebra and can be compared with the formula given at the beginning of the proof. The coefficients of ∂_{i}∂_{j} and σ_{i+j+k−p−q}∂_{i}∂_{j}∂_{k} in the two formulas do not agree as they stand, but if we symmetrize with respect to i and j (resp. to i, j and k), they do. (The somewhat tedious verification of this can be done either directly, by checking the various cases according to the inequalities satisfied by the variables, or else by forming generating functions out of the coefficients in question and comparing their values.) Similarly, if we use the commutation relations to calculate the right-hand side of (120), then after symmetrization we find the expression stated at the beginning of the proof except that the coefficient of ∂_{i}∂_{j} in the first sum has an extra term ∑_{1≤h≤p+q} ε_{p,q}^{−}(h)ε_{i,j}^{+}(h), but this sum vanishes since ε_{p,q}^{−}(h) is invariant and ε_{i,j}^{+}(h) anti-invariant under h ↦ p + q + 1 − h ( = i + j + 1 − h). Again we omit the details of the computation. □

C. Special cases and explicit formulas. In Subsection A, we gave the proof of Theorem 10 in an algebraic form that led to an explicit recursion for the coefficients, and also recast it as an explicit inductive construction of the power series G^{(n)} := ρ_{n}(G) via differential operators (eqs. (111) and (117)). Either of these can be used to make numerical calculations in any desired special case. In this final subsection we describe some of the results obtained, and a few supplementary results.

1. We start by looking at some small values of n. For the case n=1 it might seem that nothing needs to be said, since, as we have seen, the function G^{(1)}(σ_{1}) can be chosen arbitrarily. However, in fact not all choices are equally good, as we will see in a moment when we consider the next two values n = 2 and n = 3. Furthermore, there is a direct connection between the choice of G^{(1)} and the constants of proportionality relating various normalizations of our canonical bases of higher spherical polynomials, and we discuss this point briefly here because it casts some light on the calculations done in §8 for n = 3. Write the generic choice of power series G^{(1)}(σ_{1}) as

G^{(1)}(σ_{1}) = ∑_{k=0}^{∞} \frac{γ_{k}}{k!} σ_{1}^{k}  (129)

with γ_{0} = 1, where γ_{k} in general may depend on d. Our standard choice, dictated by the generating function (72) for n = 2 or (101) for n = 3, will be

γ_{k} = 2^{−k} (d − 2)_{k} ,  G^{(1)}(σ_{1}) = (1 − σ_{1}/2)^{2−d}  (130)

(or by an appropriate limiting value if d = 2). The polynomials corresponding to this normalization will be denoted simply by P_{ν ν ν}(T) (or P_{ν ν ν,d}(T)). However, we will also consider various other possible choices. The simplest and most natural one, which will occur again in points 2. and 6. below, is

\underline{γ}_{k} = 1 ,  \underline{G}^{(1)}(σ_{1}) = e^{σ_{1}} .  (131)

We will denote the corresponding generating function by \underline{G}(σ_{1}, σ_{2}, …) and its coefficients by \underline{A}(r_{1}, r_{2}, …), and notice that any other choice is related to this one by G_{k} = γ_{k} \underline{G}_{k} and A(r) = γ_{k} \underline{A}(r), where k = ∑ ir_{i}. Similarly, the higher spherical polynomials \underline{P}_{ν ν ν}(T) by

∑_{ν ν ν∈N_{0}} \underline{P}_{ν ν ν}(T) X^{ν ν ν} = \underline{G}^{(n)}(σ_{1}(XT), σ_{2}(XT), …, σ_{n}(XT)) ,  (132)

i.e., by (100) with G^{(n)} replaced by \underline{G}^{(n)}, are related to the standard polynomials by the formula

P_{ν ν ν}(T) = 2^{−k}(d − 2)_{k} \underline{P}_{ν ν ν}(T)  ( k = ∥ννν∥ := ∑_{i<j} ν_{ij} = \frac{1}{2} 1^{t} · ννν · 1 ) .  (133)

Let us look how the higher spherical polynomials defined by these various choices are related to the descending basis defined in §5. Begin with an arbitrary choice of G^{(1)} as in (129). In Theorem 10 we showed that L_{1}(G_{k}^{(n)}) = c_{k} G_{k−1}^{(n)} for certain constants c_{k} independent of n. Applying this to n = 1, with G_{k}^{(1)}(σ_{1}) = γ_{k}σ_{1}^{k}/k!, and observing that the operator L_{1} equals simply d ∂_{1} + 2σ_{1}^{2}∂_{1} for n = 1, so that L_{1}(σ_{1}^{k}) = k(d + 2 − 2)σ_{1}^{k−1}, we find that the constants c_{k} are related to the coefficients γ_{k} in (129) by c_{k} = (d + 2k − 2) γ_{k}/γ_{k−1}. In particular, for the simplest generating functions \underline{G}_{k} we have simply L_{1}(\underline{G}_{k}^{(n)}) = (d + 2k − 2) \underline{G}_{k−1}^{(n)} for all n, k ≥ 1, so from (132) and the corollary to Proposition 1 we get D_{ij}(\underline{P}_{ν ν ν}(T)) = 2(d + 2k − 2)\underline{P}_{ν ν ν−e_{ij}}(T), where k = ∥ννν∥. Comparing this with the defining property (55) of the descending basis, we obtain:

Proposition 4. The higher spherical polynomials \underline{P}_{ν ν ν}(T) defined by the generating series (132) are related to the descending basis by

\underline{P}_{ν ν ν}(T) = 2^{2k} (d/2)_{k} P_{ν ν ν}^{D}(T)  (ννν ∈ N_{0}, k = ∥ννν∥) .  (134)

Combining this with equation (133), we find that we have also proved the formula

P_{ν ν ν}(T) = 2^{k} (d/2)_{k} (d − 2)_{k} P_{ν ν ν}^{D}(T)  (ννν ∈ N_{0}, k = ∥ννν∥) .  (135)

relating our standard basis to the descending basis. This generalizes the result already proved in Proposition 3 of §8 for the case n = 3, and clarifies the origins of the two Pochhammer symbols occurring in equation (77).

2. To see which choices of γ_{k} have good properties, we look first at the case n=2, where, as we already mentioned in §8.A, the polynomials P_{ν ν ν}(T) are called Gegenbauer polynomials in general and Legendre (resp. Chebyshev) polynomials in the special case d = 3 (resp. d = 2 or 4). With the normalization fixed in (71) they are given by the generating function (72), which corresponds in our notation (since the elementary symmetric functions of the eigenvalues

of ( \begin{matrix}0 & X \\ X & 0 \end{matrix} )( \begin{matrix}m_{1} & r/2 \\ r/2 & m_{2} \end{matrix} ) are σ_{1} = rX and σ_{2} = (r^{2}/4 − m_{1}m_{2})X^{2}) to

G^{(2)}(σ_{1}, σ_{2}) = ((1 − σ_{1}/2)^{2} − σ_{2})^{−s−1}  (d = 2s + 4)  (136)

and hence to the normalization given in (130). As already stated, this will be our default choice of normalization from now on, but it seems worth mentioning that there are at least two other special choices that have special properties for n = 2 and therefore should also be considered. If we look in a standard reference work like [1], F22.9, we find, as well as the generating function (72), two other generating functions for Gegenbauer polynomials, namely, in our notations,

∑_{a=0}^{∞} \frac{2^{a}}{(2s + 2)_{a}} P_{a,d}( \begin{matrix}m_{1} & r/2 \\ r/2 & m_{2} \end{matrix} ) X^{a} = e^{rX} 𝕁_{s+1/2}((r^{2} − 4m_{1}m_{2})X^{2}) ,  (137)

∑_{a=0}^{∞} \frac{(s + 3/2)_{a}}{(2s + 2)_{a}} P_{a,d}( \begin{matrix}m_{1} & r/2 \\ r/2 & m_{2} \end{matrix} ) X^{a} = \begin{matrix}( & \frac{1 − rX/2 + \sqrt{1 − rX + m_{1}m_{2}X^{2}}}{2} & ) & −s−1/2 \\ √ & \overline{1 − rX + m_{1}m_{2}X^{2}} \end{matrix} , (138)

where 𝕁_{ν}(x) is the modified Bessel function (116). We can write these in σ_{p}-coordinates as

\begin{aligned}\underline{G}^{(2)}(σ_{1}, σ_{2}) & = e^{σ_{1}} 𝕁_{s+1/2}(4σ_{2}) , \\ Ĝ^{(2)}(σ_{1}, σ_{2}) & = \frac{1}{\sqrt{(1 − σ_{1}/2)^{2} − σ_{2}}} (\frac{1 − σ_{1}/2 + \sqrt{(1 − σ_{1}/2)^{2} − σ_{2}}}{2} )^{−s−1/2} , \end{aligned}  (139)

corresponding to the two n = 1 initializations (131) and

γ̂_{k} = (s + 1/2)_{k} ,  Ĝ^{(1)}(σ_{1}) = (1 − σ_{1}/2)^{−s−3/2} .  (140)

Notice that the generating function (138), like the standard generating function (72), is algebraic when d is rational. One can ask whether there are any other generating functions that are algebraic, at least for special values of s. In this connection we can observe that the n = 1 specializations of all three generating series in (136) and (139), namely the functions G^{(1)}, Ĝ^{(1)} and \underline{G}^{(1)} given in equations (130), (131) and (140), belong to a single one-parameter family (1−σ_{1})^{−K} (up to trivial rescalings σ_{1} ↦ λσ_{1}, with the function \underline{G}^{(1)} corresponding to the degenerate case K = λ^{−1} → ∞). The corresponding n = 2 generating series are hypergeometric, and using Schwarz’s theorem we can check that none of them is algebraic except when s is rational and K equals either 2s + 2 or s + 3/2 (corresponding to G or Ĝ) or when s is an integer, in which case they are algebraic for every value of K.

3. We next look at the case n=3, where we already know that the answer for G^{(n)} (in the standard normalization (130)) is the function given in (101). The coefficients of P_{ν ν ν}(T) as polynomials in the coefficients of T were studied in §8.D and were extremely complicated expressions that we could not evaluate completely in closed form in all cases. The coefficients of G^{(3)}, on the other hand, turn out to be very simple. For n = 3 the space W^{(3)} ⊂ K[[σ_{1}, σ_{2}, σ_{3}]] is given by the vanishing of only two operators

L_{2} = (d − 1) ∂_{2} + 2σ_{2}∂_{2}^{2} + 4σ_{3}∂_{2}∂_{3} − 2∂_{1}^{2} , L_{3} = (d − 2) ∂_{3} + 2σ_{3}∂_{3}^{2} − 4∂_{1}∂_{2} − 2σ_{1}∂_{2}^{2} .

(The other terms in (102) can be omitted by virtue of the remarks following the statement of

Proposition 1.) If we write G^{(3)} as ∑_{α,β,γ≥0} A_{α,β,γ} \frac{σ_{1}^{α}}{α!} \frac{σ_{2}^{β}}{β!} \frac{σ_{3}^{γ}}{γ!} , so that A_{α,β,γ} = A(α, β, γ) in the notation of (109), then the conditions L_{2}(G^{(3)}) = L_{3}(G^{(3)}) = 0 translate into the recursions

(d+2β+4γ−1)A_{α,β+1,γ} = 2A_{α+2,β,γ} , (d+2γ−2)A_{α,β,γ+1} = 4A_{α+1,β+1,γ} + 2αA_{α−1,β+2,γ} .

The right-hand side of the second of these equations can be replaced using the first equation by the simpler expression 2(α + 2β + 4γ + d + 1)A_{α−1,β+2,γ}, and using this one finds easily that the solution of these recursions, with the initial conditions A_{α,0,0} = 2^{−α}(2s + 2)_{α}, is given by

A_{α,β,γ} = 2^{γ−α} (2β + 4γ + 2s + 2)_{α} (2γ + s + 1)_{β} (γ + s + 1)_{γ} ,  (141)

where d = 2s + 4 and (x)_{m} = x(x + 1) ⋯ (x + m − 1) is the ascending Pochhammer symbol as usual. We thus obtain the following very simple description of G^{(3)} as a power series.

Proposition 5. The generating function G^{(3)} is given by

G^{(3)}(σ_{1}, σ_{2}, σ_{3}) = ∑_{α, β, γ ≥ 0} 2^{γ−α} \binom{α + 2β + 4γ + 2s + 1}{α}\binom{β + 2γ + s}{β}\binom{2γ + s}{γ} σ_{1}^{α}σ_{2}^{β}σ_{3}^{γ} ,

(Note that we could also have obtained this expansion directly, using equation (70) and the binomial theorem, as the Taylor expansion of the algebraic function (101).)

We also mention that the n = 3 case allows us to reduce the freedom of choice that we still had for n = 2, since neither the function Ĝ^{(3)} corresponding to the functions (137) or (140), nor the functions n = 3 corresponding to G^{(1)}(σ_{1}) = (1−σ_{1})^{−K} for s integral and other choices than K = s + 3/2 or K = 2s + 2, turn out to be algebraic, leaving out standard normalization (130) as the only one (up to rescaling) that gives algebraic generating functions up to n = 3. From now on we always assume that we have made this choice.

4. For n> > >3 the coefficients of the generating function G^{(n)} no longer factor nicely into linear factors, and the generating function itself no longer seems to be algebraic in general for any choice of the initial function G^{(1)}. We will consider in detail the case n=4. Write G^{(4)} = ∑_{α,β,γ,δ≥0} A_{α,β,γ,δ} \frac{σ_{1}^{α}}{α!} \frac{σ_{2}^{β}}{β!} \frac{σ_{3}^{γ}}{γ!} \frac{σ_{4}^{δ}}{δ!} . Then the equation L_{4}(G^{(4)}) = 0 gives the recursion

\frac{d + 2δ − 3}{2} A_{α,β,γ,δ+1} = 2A_{α+1,β,γ+1,δ} + A_{α,β+2,γ,δ} + 2αA_{α−1,β+1,γ+1,δ} + βA_{α,β−1,γ+2,δ} ,

and this allows us to reduce the calculation of all A_{α,β,γ,δ} to that of A_{α,β,γ,0} = A_{α,β,γ}, which is given by (141). Already for δ = 1 the explicit formulas for the coefficients A_{α,β,γ,δ} become complicated, but they can nevertheless be used to calculate as many terms of the power series G^{(4)} as we want, or alternatively, we can use eq. (115) to compute G^{(4)} directly from G^{(3)}. Carrying out these calculations for various small values of d (d = 4, 5, …, 10) and up to fairly large degrees led to a number of conjectural statements, most of which we were then able to prove. These experiments, results, and proofs will be described in detail in §10. Here we mention only the highlights:

(a) For d = 4 the function G^{(4)} is algebraic.

(b) For d = 6, 8 and 10 the function G^{(4)} is not algebraic, but certain derivatives of it are. Specifically, the images of G^{(4)} under E + 3 for d = 6, under (E + 4)(E + 5)(E + 6) for d = 8, and under (E + 5)(E + 6)(E + 7)(E + 8)(E + 9) for d = 10, where E as usual denotes the Euler operator, all turned out experimentally to be algebraic functions, leading to the conjectural statement that \binom{E+3s}{2s−1}G^{(4)}(σ_{1}, σ_{2}, σ_{3}, σ_{4}; 2s + 4) is algebraic for all positive integral values of s. This will be proved in §10.

(c) For d = 5, 7 and 9 we find numerically that the power series G^{(4)} is not algebraic, but does have integral coefficients after rescaling by some simple factor (involving only the prime 2 for d = 5 and d = 7 and only the primes 2 and 3 for d = 9). We conjecture that this is true for all odd values of d ≥ 5.

5. In points 1.–4. above, we looked at the full generating series G^{(n)} for small values of n. In a different direction, we can consider instead small values of the weight k. Here we can give a closed formula for any fixed value of k, the first examples being

G_{1} = \frac{1}{2} (d − 2) σ_{1} ,
G_{2} = \frac{1}{4} \binom{d − 1}{2} σ_{1}^{2} + \frac{d − 2}{2} σ_{2} ,
G_{3} = \frac{1}{8} \binom{d}{3} σ_{1}^{3} + \frac{d(d − 2)}{4} σ_{1}σ_{2} + d σ_{3} ,
G_{4} = \frac{1}{16} \binom{d + 1}{4} σ_{1}^{4} + \frac{(d + 1)d(d − 2)}{16} σ_{1}^{2}σ_{2} + \frac{d(d + 2)}{2} σ_{1}σ_{3} + \frac{d(d − 2)}{8} σ_{2}^{2} + \frac{d(5d + 6)}{2(d − 3)} σ_{4} .

Expressed differently, for each value of k there are only finitely many coefficients A(r_{1}, r_{2}, …) of weight k, corresponding to the partitions k = r_{1} + 2r_{2} + ⋯, and each of these coefficients is a rational function of s independent of n, e.g., the coefficient A(3, 2, 0, 1) of σ_{1}^{3}σ_{2}^{2}σ_{4}/12 (of total weight k = 11) is given by

A(3, 2, 0, 1; d) = \frac{(s + 3)(s + 4)(s + 5)(s + 6)(10s^{3} + 169s^{2} + 935s + 1634)}{2s + 1} ,

where we have written d = 2s + 4 as usual. In general these rational functions are complicated, as this example shows, and it is unlikely that one give a general formula for them in closed form. However, we can give a complete formula for their asymptotic behavior as s (or d) tends to infinity, e.g. A(3, 2, 0, 1; d) ∼ 5s^{6} as s → ∞ in the above example.

Proposition 6. For each fixed value of r = (r_{1}, r_{2}, …), the coefficient A(r; d = 2s + 4) of ∏_{p} σ_{p}^{r_{p}}/r_{p}! in G(σ_{1}, σ_{2}, …) is a rational function of s with asymptotic behavior

A(r; 2s + 4) ∼ (1^{r_{1}} 1^{r_{2}} 2^{r_{3}} 5^{r_{4}} ⋯) s^{r_{1}+r_{2}+r_{3}+···} ,  (142)

as s → ∞, where the coefficient is ∏_{i=1}^{∞} C_{i−1}^{r_{i}} with C_{n} = \frac{1}{n+1} \binom{2n}{n}, the nth Catalan number.

Proof. We already know that A(r; d) is a rational function of s (in fact, an element of the ring R defined in (112)) and that it is determined completely by the recursion (111) with the initial conditions A(k, 0, 0, …; d) = γ_{k}, with γ_{k} as in (130). In this recursion each coefficient A(r) is expressed as a linear combination of coefficients A(r′′′) with r′_{i} = r_{i} − δ_{i,p} + δ_{i,a} + δ_{i,b} − δ_{i,a+b−p} for some integers a, b > 0 with a + b ≥ p. Since each r′′′ is less than r in lexicographical ordering, we may assume by induction that (142) holds for A(r′′′; d). (The initial value of the induction is correct, since γ_{k} = 2^{−k}(d − 2)_{k} ∼ s^{k} as s → ∞.) If a + b > p, then ∑_{i≥1} r′_{i} = ∑_{i≥1} r_{i}, so A(r′′′; d) grows like the same power of s as in the as-yet-unproved formula for A(r; d), and the factor 2/(d − p + 2r_{p} − 1) ∼ 1/s in front of the sum in (111) implies that these terms give a negligible asymptotic contribution. If a + b = p, then ∑_{i≥1} r′_{i} is larger by 1 than ∑_{i≥1} r_{i}, so the contributions of these terms in (130) have the same power of s as in (111). Also, the product ∏_{i≥1} C_{i−1}^{r′_{i}} is equal to C_{a−1}C_{b−1}/C_{p−1} times the product ∏_{i≥1} C_{i−1}^{r_{i}}, so the correctness of formula (142) follows from the identity ∑_{a+b=p} C_{a−1}C_{b−1} = C_{p−1}, which is the standard recursion relation for the Catalan numbers. □

6. In fact we can extend the idea of 5. further by giving closed formulas for A(r; d) when only r_{2}, r_{3}, … are fixed and r_{1} (or equivalently k = r_{1} + 2r_{2} + 3r_{3} + ⋯) as well as d is allowed to vary. For instance, we find that for every value of k and n the coefficient A(k − 2, 1, 0, …) of σ_{1}^{k−2}σ_{2}/(k − 2)! in G_{k}^{(n)} is equal to 2^{1−k}(d − 2)_{k}/(d − 1). The factor of 2^{−k}(d − 2)_{k} in this expression suggests that it would be better to work with the other normalization \underline{A}(r) (cf. (133)), in which case the formula in question simplifies to just \underline{A}(k − 2, 1, 0, …) = 2/(d − 1). In this example the value is independent of k and is an element of the ring R defined in (112), but in general it turns out to be a polynomial in k with coefficients in R. In other words, we have an expansion of the weight k part of \underline{G} as

\underline{G}_{k}(σ_{1}, σ_{2}, …, σ_{k}) = ∑_{l=0}^{k} H_{l}(σ_{2}, σ_{3}, …, σ_{l}; k) \frac{σ_{1}^{k−l}}{(k − l)!}  (143)

where each H_{l} is a polynomial of (weighted) degree l in σ_{2}, …, σ_{l} with coefficients in R[k], the first values being

H_{0} = 1 , H_{1} = 0 , H_{2} = \frac{2}{d − 1} σ_{2} , H_{3} = 8 \frac{d + k − 2}{(d + 1)(d − 1)(d − 2)} σ_{3} ,
H_{4} = 8 \frac{5d^{2} + (12k − 27)d + 8k^{2} − 36k + 34}{(d + 3)(d + 1)(d − 1)(d − 2)(d − 3)} σ_{4} + \frac{4}{(d + 1)(d − 1)} \frac{σ_{2}^{2}}{2!} ,
H_{5} = 32 \frac{P_{5}}{(d + 5)(d + 3)(d + 1)d(d − 1)(d − 2)(d − 3)(d − 4)} σ_{5}

+ 16\frac{d + k − 2}{(d + 3)(d + 1)(d − 1)(d − 2)} σ_{2}σ_{3} ,

H_{6} = 64 \frac{P_{6}}{(d + 7)(d + 5)(d + 3)(d + 1)d(d − 1)(d − 2)(d − 3)(d − 4)(d − 5)} σ_{6}

+ 16 \frac{5d^{3} + (12k − 25)d^{2} + (8k^{2} − 28k + 22)d + 8(k − 1)(k − 2)}{(d + 5)(d + 3)(d + 1)d(d − 1)(d − 2)(d − 3)} σ_{2}σ_{4}

+ 64 \frac{(d + k − 1)(d + k − 2)}{(d + 5)(d + 3)(d + 1)d(d − 1)(d − 2)} \frac{σ_{3}^{2}}{2!} + \frac{8}{(d + 3)(d + 1)(d − 1)} \frac{σ_{2}^{3}}{3!}

with

P_{5} = 7d^{4} + (28k − 72)d^{3} + (40k^{2} − 206k + 231)d^{2} + (20k^{3} − 152k^{2} + 338k − 210)d
+ 4(k − 1)(k − 2)(k − 5) ,
P_{6} = 21d^{5} + (120k − 350)d^{4} + (270k^{2} − 1574k + 2047)d^{3} + (280k^{3} − 2412k^{2} + 6188k − 4582)d^{2}
+ (112k^{4} − 1200k^{3} + 4166k^{2} − 5278k + 1760)d + 40(k − 1)(k − 2)(k − 6)(2k − 7) .

Here we should warn the reader that, although all the coefficients shown in this list have only simple poles in d, this property fails in general and in fact already in degree 7, where the coefficient of σ_{2}σ_{5} \frac{σ_{1}^{k−7}}{(k−7)!} has a factor (d+1)^{2} in its denominator. (That this example involves σ_{5} is no accident, since it follows immediately from (115) that A(r_{1}, r_{2}, r_{3}, r_{4}) has a denominator at most (s + 1/2)_{r_{4}} and hence has at most simple poles.) It is also perhaps worth mentioning that formula (143) for the weight k part \underline{G}_{k} of the generating function can also be written in an equivalent form for the full function \underline{G} = ∑_{k} \underline{G}_{k} as

e^{−σ_{1}} \underline{G}(σ_{1}, σ_{2}, σ_{3}, …) = 1 + \frac{2}{d − 1} σ_{2} + 8 \frac{d + 1 + σ_{1}}{(d + 1)(d − 1)(d − 2)} σ_{3}

+ 8 \frac{5d^{2} + 21d + 18 + 12(d + 3)σ_{1} + 8σ_{1}^{2}}{(d + 3)(d + 1)(d − 1)(d − 2)(d − 3)} σ_{4} + \frac{4}{(d + 1)(d − 1)} \frac{σ_{2}^{2}}{2!} + ⋯

∈ R[σ_{1}][[σ_{2}, σ_{3}, …]] .  (144)

7. Our next point is that the new expansion (143) has an asymptotic property generalizing Proposition 6 to the case when k and s both go to infinity. Notice first that equation (142), when written in terms of the renormalized coefficients \underline{A}(r; d), takes the form

\underline{A}(r; 2s + 4) ∼ (1^{r_{2}} 2^{r_{3}} 5^{r_{4}} ⋯) s^{−r_{2}−2r_{3}−3r_{4}−···} ,

in which r_{1} does not occur at all. This is for k constant, but suggests now also looking at the asymptotics for s → ∞ of the coefficients of \underline{A}(r; 2s+4) as polynomials in k. If we do this for the values listed in the table of H_{l} above, then we see a nice multiplicative property: the coefficient of σ_{2} in H_{2} is asymptotically equal to \frac{1}{s} as s → ∞, the coefficient of σ_{4} in H_{4} is asymptotically equal to \frac{5s^{2}+6ks+2k^{2}}{s^{5}} as s → ∞, and the coefficient of σ_{2}σ_{4} in H_{6} is asymptotically equal to \frac{5s^{2}+6ks+2k^{2}}{s^{6}} , which is the product of these two expressions. Similarly, the coefficient of σ_{3}^{2}/2! in H_{6} is asymptotically equal to \frac{(2s+k)^{2}}{s^{6}} as s → ∞, and this is the square of the asymptotic formula for the coefficient of σ_{3} in H_{3}. In other words, we seem to have an asymptotic formula

\underline{A}(r; 2s + 4) ∼ ∏_{p=1}^{∞}(\frac{c_{p−1}(k/s)}{s^{p−1}})^{r_{p}}  (145)

as s and k go simultaneously to infinity, where the c_{n}(λ) are certain universal polynomials of λ = k/s whose first few values are

c_{0}(λ) = 1,  c_{1}(λ) = 1,  c_{2}(λ) = λ + 2,  c_{3}(λ) = 2λ^{2} + 6λ + 5,
c_{4}(λ) = 5λ^{3} + 20λ^{2} + 28λ + 14, c_{5}(λ) = 14λ^{4} + 70λ^{3} + 135λ^{2} + 120λ + 42 .

(In particular, the formula (145) remains unchanged if one replaces the product by one starting at p = 2 instead of p = 1.) To guess what these polynomials are, we first notice that compatibility with Proposition 6 requires that the constant term c_{n}(0) equals C_{n}, the nth Catalan number, and that by inspection the highest term of c_{n}(λ) for n ≥ 1 is C_{n−1}λ^{n−1}. Since the generating function of ∑_{n≥0} C_{n}x^{n} is the reciprocal of a slightly simpler function, we look at the first few terms of the reciprocal of the series ∑_{n≥0} c_{n}(λ)x^{n} and immediately find that the coefficient of x^{n} in this reciprocal has the very simple form −C_{n−1}(λ + 1)^{n−1} for all n ≥ 1. This leads to the definition of the polynomials c_{n}(λ) given in the following proposition, whose proof will be shorter than the discussion leading up to it.

Proposition 7. For fixed values of r_{2}, r_{3}, ⋯ ≥ 0, the coefficient \underline{A}(r_{1}, r_{2}, r_{3}, …; d = 2s + 4) of ∏_{p≥1} σ_{p}^{r_{p}}/r_{p}! in G_{k}(σ_{1}, σ_{2}, …) (k = r_{1} + 2r_{2} + ⋯) is a polynomial in r_{1} or in k of degree ∑_{p≥3}(p − 2)r_{p} with coefficients in the ring R, and the asymptotic behavior of this coefficient as s and k tend to infinity is given by (145), where the polynomials c_{n}(λ) are defined by the closed formula

c_{n}(λ) = \{ \begin{matrix}1 & if n = 0, \\ \frac{(2n)!}{n!(n − 1)!} ∑_{j=0}^{n−1} \binom{n − 1}{j} \frac{λ^{j}}{(n + j)(n + j + 1)} & if n ≥ 1 \end{matrix}  (146)

or by the generating function

∑_{n=0}^{∞} c_{n}(λ) x^{n} = \frac{1 + 2λ − \sqrt{1 − 4(λ + 1)x}}{2 (λ + x)} .  (147)

Proof. Both the general form (143) of the generating function and the formula (145) for the asymptotic behavior of its coefficients follow from the recursion relation (111) for these coefficients (which is equally valid for the coefficients \underline{A}(r) as for the standard coefficients A(r), only the initial conditions being different), and the proof of Proposition 7 is almost identical to the proof of Proposition 6: the only difference is that the terms in (111) with a + b = p + 1 now also contribute to the leading term of the expansion (because the coefficient r_{1} of A(r′′′) for these terms is of the same order as s) and that the identity C_{p−1} = ∑_{a+b=p} C_{a−1}C_{b−1} used in the proof of Proposition 6 must therefore be replaced by the more complicated identity

c_{p−1}(λ) = ∑_{\begin{matrix}a, b≥1 \\ a+b=p \end{matrix}} c_{a−1}(λ) c_{b−1}(λ) + λ ∑_{\begin{matrix}a, b≥2 \\ a+b=p+1 \end{matrix}} c_{a−1}(λ) c_{b−1}(λ) (p ≥ 2) ,

which is in turn easily seen to be equivalent to the generating function (147). □

8. Finally, we can use our generating functions to prove two statements about the determinant of the Gram matrices discussed in §5. We showed there (Theorem 6) that the monomial and descending bases are dual to one another with respect to the scalar product of §3, and hence that the two N_{0}(a) × N_{0}(a) Gram matrices

(P_{μ μ μ}^{M}, P_{ν ν ν}^{M})_{μ μ μ, ν ν ν∈N_{0}(a)}  and  (P_{μ μ μ}^{D}, P_{ν ν ν}^{D})_{μ μ μ, ν ν ν∈N_{0}(a)}

are inverses of one another for generic values of d. From Theorem 11 we know that both matrices are regular and invertible away from integer values of d ≤ n − 1. It follows that the “Gram determinant” function GD_{a}^{M}(d) ∈ ℚ(d) defined by

GD_{a}^{M}(d) = det( (P_{μ μ μ}^{M}, P_{ν ν ν}^{M})_{μ μ μ, ν ν ν∈N_{0}(a)})

is a rational function with zeros and poles only at integer values of its argument, i.e., that GD_{a}^{M}(d) has the form C ∏_{m}(d+m)^{e(m)} for some constant C = C_{a} and exponents e(m) = e_{a}(m) depending on a, with e(m) = 0 for all but finitely many values of m, and e(m) = 0 for all m ≤ −n. In the next proposition we will (1) give C and ∑_{m} e(m) (or equivalently, the asymptotics of GD_{a}^{M}(d) as |d| → ∞) in all cases, and (2) give a complete formula for GD_{a}^{M}(d) whenever n > 3 and N_{0}(a) = 1 (so that GD_{a}^{M}(d) is the determinant of a 1 × 1 matrix), in which case the exponents e(m) turn out to depend only on the half-degree k = \frac{1}{2}(a_{1} + ⋯ + a_{n}).

Proposition 8. (1) As d → ∞ with a fixed, we have

GD_{a}^{M}(d) ∼ ∏_{ν ν ν∈N_{0}(a)}(ννν!!! d^{k})

where ννν!!! := ∏_{i<j} ν_{ij}! .

(2) Suppose that N_{0}(a) = 1 for some a with n ≥ 4 and a_{i} ≥ 1 for all i. Then

GD_{a}^{M}(d) = ( ∏_{i=1}^{n−1} a_{i}!) (d + 2k − 2) (d − 1)_{k−1} .

Proof. (1) As in 5. above, we denote by G_{k} the weight k part of the standard generating function. Then by (117), we have

G_{k} = \frac{(d − 2)_{k}}{2^{k}k!} σ_{1}^{k} + O(d^{k−1}) .

This is because the operator σ_{i}M_{i} with 2 ≤ i ≤ n on polynomials of weight k in σ_{1}, …, σ_{i−1} does not change the weight and the action of ((d − i + 1)/2)_{r}(σ_{i}M_{i})^{r} decreases the order by d^{r}, so the dominant asymptotic term as d → ∞ comes from the coefficient of σ_{1}^{k}. For any index ννν of weight k, the coefficient of x^{ν ν ν} in σ_{1}^{k} is given by (2^{k}k!/ννν!) T^{ν ν ν}, so by (135) we have

P_{ν ν ν}^{D} = \frac{1}{ννν! (d/2)_{k}} T^{ν ν ν} + O(d^{−k−1}) = \frac{1}{ννν! (d/2)_{k}} P_{ν ν ν}^{M} + O(d^{−k−1})

and hence

(P_{μ μ μ}^{D}, P_{ν ν ν}^{D}) ∼ \frac{δ_{μ μ μ,ν ν ν}}{ννν! d^{k}} + O(d^{−k−1}) .

Since GD_{a}^{M}(d) = det((P_{μ μ μ}^{D}, P_{ν ν ν}^{D}))^{−1}, we have the assertion.

(2) By symmetry, we may assume that a_{1} ≤ a_{2} ≤ ⋯ ≤ a_{n}. Since N_{0}(a) = 1, we have a_{n} = a_{1} + ⋯ + a_{n−1} (cf. Remark 2. in §2) and the corresponding index ννν = (ν_{ij}) ∈ N_{0}(a) satisfies ν_{in} = a_{i} for i < n and ν_{ij} = 0 for other (i, j). In order to calculate GD_{a}^{M}(d), we only need the coefficient of P_{ν ν ν}^{D} at T^{ν ν ν}. We see this coefficient by the generating function. To see this, since ν_{ij} = 0 if i < n and j < n, we may put x_{ij} = 0 for pairs (i, j) with i < n and j < n in the generating function. Under this substitution, we have σ_{l} = 0 for l ≥ 3, σ_{1} = 2 ∑_{i=1}^{n−1} t_{in}x_{in} and σ_{2} =( ∑_{i=1}^{n−1} t_{in}x_{in})^{2} = σ_{1}^{2}/4. So the standard generating function reduces to

((1 − σ_{1}/2)^{2} − σ_{2})^{−(d−2)/2} = (1 − σ_{1})^{−(d−2)/2} = ∑_{k=0}^{∞} \frac{((d − 2)/2)_{k}}{k!} σ_{1}^{k} .

Expanding σ_{1}^{k}, the coefficient of x^{ν ν ν} = x_{1n}^{a_{1}} ⋯ x_{n−1,n}^{a_{n−1}} is given by 2^{k} · ((d − 2)/2)_{k}/a_{1}! ⋯ a_{n−1}!. By (135), we have

(P_{ν ν ν}^{D}, P_{ν ν ν}^{D}) = \frac{1}{(d − 1)_{k−1}(d + 2k − 2) a_{1}! ⋯ a_{n−1}!} .

This proves the assertion. □

§10. Algebraic generating functions for the case n = 4

As mentioned in point C.4. of §9, the formalism developed in that section permitted extensive numerical calculations of the generating function G^{(4)}(σ_{1}, σ_{2}, σ_{3}, σ_{4}) that led to the formulation of a number of interesting properties, some of which we later proved and some of which remain conjectural. In this section give these results and their proofs, also including a brief description of the experiments that led up to them, without which the statements would be completely unmotivated. The first three subsections treat the case of even values of d, proving the algebraicity of \binom{E+3s}{2s−1}G^{(4)} (where d = 2s + 4 as usual) that was already mentioned in §9.C.4. In §10.D we strengthen this result by proving that the generating series having these functions as coefficients is also algebraic. The last two subsections are devoted to a brief discussion of the case of odd d, for which the results are still conjectural, and to a final remark concerning the case when n is larger than 4.

A. Even values of d: experimental results. We look first at the case d = 4 (s = 0), since the generating function for n = 3 was particularly simple for this value of d (§8.A). The expansion of G^{(4)}(σ_{1}, …, σ_{4}) for d = 4 begins

1 + σ_{1} + \frac{3σ_{1}^{2} + 4σ_{2}}{4} + \frac{σ_{1}^{3} + 4σ_{1}σ_{2} + 8σ_{3}}{2} + \frac{5σ_{1}^{4} + 40σ_{1}^{2}σ_{2} + 192σ_{1}σ_{3} + 16σ_{2}^{2} + 832σ_{4}}{16} + · · · .

To simplify the coefficients of this and thus improve our chances of recognizing the function, we specialize the variables σ_{p}. Since we already know the n = 3 function, we must not specialize σ_{4} to 0, so the simplest choice is to take σ_{1} = σ_{2} = σ_{3} = 0. Doing this, we find that the function y = G^{(4)}(0, 0, 0, x/4) (where the factor 1/4 is included just to reduce the size of the coefficients) has a power series expansion in x beginning

y = 1 + 13 x + 321 x^{2} + 8989 x^{3} + 265729 x^{4} + 8097453 x^{5} + · · · .

Calculating this series to many terms and looking for linear relations among monomials of small degree in x and y, we find that within the accuracy of the calculation the function y satisfies the quadratic equation

(1 − 34x + x^{2})^{2} y^{4} − (1 + x)(1 − 34x + x^{2}) y^{2} + 9x = 0 ,

the solution of which is the explicit algebraic function

y = \sqrt{\frac{1 + x + \sqrt{1 − 34x + x^{2}}}{2 (1 − 34x + x^{2})}} = \frac{1}{2\sqrt{1 − 6\sqrt{x} + x}} + \frac{1}{2\sqrt{1 + 6\sqrt{x} + x}} .  (148)

Repeating the calculation for various other specializations of G^{(4)} and interpolating the results obtained, we are led to conjecture the following result, whose proof will be given in §10.B.

Proposition 1. The function y_{0} = y_{0}(σ_{1}, σ_{2}, σ_{3}, σ_{4}) = G^{(4)}(σ_{1}, σ_{2}, σ_{3}, σ_{4}; d = 4) is algebraic of degree 4 over ℚ(σ_{1}, …, σ_{4}) and is given explicitly by

y_{0} = \sqrt{\frac{Δ_{1} + \sqrt{Δ_{2}}}{2 Δ_{2}}} ,  (149)

where Δ_{1} and Δ_{2} are defined by

Δ_{0} = (1 − σ_{1}/2)^{2} − σ_{2} ,  Δ_{1} = Δ_{0}^{2} − 8σ_{3} + 4σ_{4} ,  Δ_{2} = Δ_{1}^{2} − 16σ_{4}(4 − 2σ_{1} − Δ_{0})^{2} .

We remark that for σ_{4} = 0 we have Δ_{1} = Δ_{0}^{2} −8σ_{3} and Δ_{2} = Δ_{1}^{2}, so the right-hand side of (149) reduces in that case to 1/\sqrt{Δ_{0}^{2} − 8σ_{3}}, in accordance with formula (101) when d = 4.

After this initial success we look at the case d = 5, but here there seems to be no algebraic formula. (We will return to this question in §10.E.) On the other hand, when we look at the following case d = 6, the situation is better. Here the expansion of y = G^{(4)}(0, 0, 0, x/4) begins

1 + 9 x + 153 x^{2} + \frac{16213}{5} x^{3} + ⋯ + \frac{24075859253492406153684681487509}{29} x^{21} + · · · .

Unlike the corresponding series for d = 4 and d = 5, which to the accuracy of the computation were in ℤ[[x]] and ℤ[[x/64]], respectively, this series appears to have infinitely many primes in its denominator, in which case it is definitely not algebraic since the Taylor coefficients of an algebraic function always belong to ℤ[1/N] for some integer N. On the other hand, looking more carefully, we find that the denominator of the coefficient of x^{n} is always a divisor of 4n+3, which means that the power series y_{1} := 4xy′ + 3y has integral coefficients, and now applying the same numerical procedure as before to y_{1} leads to the experimental formula

y_{1} = \sqrt{\frac{1 + x − \sqrt{1 − 34x + x^{2}}}{2 x (1 − 34x + x^{2})}} = \frac{1}{2\sqrt{x(1 − 6\sqrt{x} + x)}} − \frac{1}{2\sqrt{x(1 + 6\sqrt{x} + x)}} , (150)

very similar to the corresponding formula (148) for the d = 4 case. More generally, when we look at the Taylor expansion of y = G^{(4)}(σ_{1}, σ_{2}, σ_{3}, σ_{4}; 6) for general values of the σ_{p}, we find that the denominators in the coefficients are always divisors of k + 3, where k is the degree of the monomial in question, so that we must replace y by y_{1} = (E + 3)y to get integrality, where E = ∑_{p} pσ_{p} is the Euler operator that multiplies a monomial of weight k in the σ_{p} by k. Just as before, by identifying the power series obtained for several other specializations as algebraic functions and interpolating, we are led to guess the following proposition, whose proof will also be given in Subsection B below.

Proposition 2. The generating function G^{(4)} is not algebraic for d = 6, but its image under E + 3, where E is the Euler operator defined above, is algebraic and is given by the formula

y_{1} = \sqrt{\frac{Δ_{1} − \sqrt{Δ_{2}}}{8 σ_{4} Δ_{2}}} ,  (151)

where Δ_{1} and Δ_{2} have the same meaning as in Proposition 1.

We observe that the product of y_{1} and the function y_{0} in (149) equals (4 − 2σ_{1} − Δ_{0})/Δ_{2}, which is a rational function in the initial variables σ_{p}, so that the function fields generated by the two algebraic functions y_{0} and y_{1} coincide.

Continuing to higher values, we find that for d = 8 (s = 2) we have to apply the differential operator \frac{1}{6}(E + 4)(E + 5)(E + 6) = \binom{E+6}{3} to y to achieve integrality and that it is then algebraic (this case will also be discussed further in §10.B), while for d = 10 (s = 3) we need to apply \binom{E+9}{5} to get integrality of the Taylor coefficients, with no smaller polynomial in E sufficing. These special cases suggest the following theorem, whose proof will occupy most of this section.

Theorem 13. For s ∈ ℤ_{≥0} define y_{s} = y_{s}(σ_{1}, σ_{2}, σ_{3}, σ_{4}) by

y_{s} = \{ \begin{matrix}G^{(4)}(σ_{1}, σ_{2}, σ_{3}, σ_{4}; 4) & if s = 0, \\ \binom{E+3s}{2s−1}G^{(4)}(σ_{1}, σ_{2}, σ_{3}, σ_{4}; 2s + 4) & if s ≥ 1. \end{matrix}  (152)

Then each y_{s} is algebraic of degree 4 over K = ℚ(σ_{1}, σ_{2}, σ_{3}, σ_{4}). More precisely, each y_{s} belongs to the 2-dimensional subspace Ky_{0}+Ky_{1} = K(\sqrt{Δ_{2}}) y_{0} of the degree 4 extension K(y_{0}) = K(y_{1}) of K, where y_{0} and y_{1} are given by (149) and (151), respectively.

Remark. Applying a polynomial P (E) in the Euler operator to the generating function G^{(n)} is the same as multiplying the degree k terms by P (k) for each k ≥ 0, and hence is equivalent to working with a different initialization G^{(1)}(σ_{1}) = ∑_{k≥0} P (k)(d − 2)_{k}(σ_{1}/2)^{k}/k! of the n = 1 generating function. So our discussion can be summarized by saying that for s ∈ ℤ_{≥0} there is a normalization of the initial generating function G^{(1)} which makes G^{(n)} algebraic for all n ≤ 4.

Finally, we mention an identity which is of interest in itself and will have an interpretation in terms of the generating series with respect to s studied in §10.D. For s ∈ ℕ, define

y_{s}^{±}(0) = y_{s}^{±}(σ_{1}, σ_{2}, σ_{3}, 0) = \binom{E + 3s}{2s − 1}(\frac{A_{±}^{s}}{\sqrt{Δ_{0}^{2} − 8σ_{3}}} ) ,  (153)

where A_{±} := (Δ_{0} ± \sqrt{Δ_{0}^{2} − 8σ_{3}})/2 , and

y_{s}^{±} = y_{s}^{±}(σ_{1}, σ_{2}, σ_{3}, σ_{4}) = ∑_{r=0}^{∞} M_{4}^{r}(y_{s}^{±}(σ_{1}, σ_{2}, σ_{3}, 0)) \frac{σ_{4}^{r}}{r! (s + \frac{1}{2})_{r}} ,  (154)

so that (by (115)) y_{s} = y_{s}^{+}. The following identity will be proved in §10.C.

Theorem 14. The functions y_{s}^{+} and y_{s}^{−} coincide for s ∈ ℕ.

B. Algebraicity and non-algebraicity proofs for G^{(4)} when s is integral. In this section we give the proofs of Propositions 1 and 2 and begin the discussion of higher even values of d. These proofs involve the introduction of new coordinates for the ring ℚ[σ_{1}, σ_{2}, σ_{3}, σ_{4}] and thus suggest the problem, which may be an interesting direction for further research, of finding new coordinates for the whole ring ℚ[σ_{1}, σ_{2}, …] that could help to clarify the structure of the generating function G^{(n)} for higher values of n. (We return to this point briefly in §10.F.)

Motivated by the experimental formula (149) and a certain amount of trial and error, we make the (invertible) change of variables

σ_{1} = 2u_{1} + 2 ,  σ_{2} = u_{2} + u_{1}^{2} + 4u_{1} ,  σ_{3} = u_{3} + u_{1}u_{2} + 2u_{1}^{2} .  (155)

In terms of these variables the operator M_{4} defined in (113) becomes simply

M_{4} = \frac{∂^{2}}{∂u_{2}^{2}} + \frac{∂^{2}}{∂u_{1}∂u_{3}} ,  (156)

and, if we write g_{3} and g_{4} for the functions G^{(3)} and G^{(4)} expressed in the variables u_{i} and σ_{4}, then equations (101) and (115) take the form

g_{3}(u_{1}, u_{2}, u_{3}; d = 2s + 4) = \frac{1}{\sqrt{u_{2}^{2} − 8u_{3}}} (\frac{−4u_{1} − u_{2} + \sqrt{u_{2}^{2} − 8u_{3}}}{2} )^{−s} ,  (157)

g_{4}(u_{1}, u_{2}, u_{3}, σ_{4}; d) = ∑_{r=0}^{∞}(\frac{∂^{2}}{∂u_{2}^{2}} + \frac{∂^{2}}{∂u_{1}∂u_{3}} )^{r}(g_{3}(u_{1}, u_{2}, u_{3}; d)) \frac{σ_{4}^{r}}{r! (s + \frac{1}{2})_{r}} .  (158)

If s = 0, then g_{3} is independent of u_{1} and (158) simplifies to g_{4} = ∑_{r=0}^{∞} \frac{∂^{2r}g_{3}}{∂u_{2}^{2r}} \frac{σ_{4}^{r}}{r! (1/2)_{r}} . This

formula makes it very natural to make the further (non-invertible) change of variables

σ_{4} = X^{2}/4 ,  (159)

in which case it can be written using Taylor’s formula and the duplication formula for the gamma function as

g_{4}(u_{1}, u_{2}, u_{3}, X^{2}/4) = ∑_{r=0}^{∞} \frac{∂^{2r}g_{3}(u_{1}, u_{2}, u_{3})}{∂u_{2}^{2r}} \frac{X^{2r}}{(2r)!} = \frac{g_{3}(u_{1}, u_{2} + X, u_{3}) + g_{3}(u_{1}, u_{2} − X, u_{3})}{2}

= \frac{1}{2} (\frac{1}{\sqrt{(u_{2} + X)^{2} − 8u_{3}}} + \frac{1}{\sqrt{(u_{2} − X)^{2} − 8u_{3}}} ) .  (160)

Squaring this gives

g_{4}^{2} = \frac{u_{2}^{2} + X^{2} − 8u_{3} + \sqrt{(u_{2}^{2} + X^{2} − 8u_{3})^{2} − 4u_{2}^{2}X^{2}}}{2((u_{2}^{2} + X^{2} − 8u_{3})^{2} − 4u_{2}^{2}X^{2})} = \frac{Δ_{1} + \sqrt{Δ_{2}}}{2Δ_{2}}

with Δ_{1} and Δ_{2} as in Proposition 1, completing the proof of equation (149).

The case s = 1 (d = 6) is only slightly more complicated. Write y^{0} for the expression (101) with s = 1 and y_{1}^{0} for its image under E + 3, where E denotes the Euler operator ∑ p σ_{p} ∂/∂σ_{p}. Because E is a derivation and E(σ_{3}) = 3σ_{3}, we find

y_{1}^{0} = (E + 3)[\frac{1}{4σ_{3}} (\frac{Δ_{0}}{\sqrt{Δ_{0}^{2} − 8σ_{3}}} − 1)] = \frac{1}{4σ_{3}} E(\frac{Δ_{0}}{\sqrt{Δ_{0}^{2} − 8σ_{3}}} )

= \frac{−2E(Δ_{0}) + 3Δ_{0}}{(Δ_{0}^{2} − 8σ_{3})^{3/2}} = \frac{u_{2}}{(u_{2}^{2} − 8u_{3})^{3/2}} = \frac{∂}{∂u_{2}} (\frac{−1}{\sqrt{u_{2}^{2} − 8u_{3}}} ) ,

which (a) is again independent of u_{1}, and (b) can be expressed as a u_{2}-derivative. Now the relationship between y_{1} = (E + 3)G^{(4)} and y_{1}^{0} is the same as the relationship (115) between y = G^{(4)} and y^{0}, because the operator σ_{4}M_{4} (or σ_{n}M_{n} for any n) is homogeneous of degree 0 and therefore commutes with E. We therefore get

y_{1} = ∑_{r=0}^{∞} \frac{∂^{2r}}{∂u_{2}^{2r}} (y_{1}^{0}) \frac{σ_{4}^{r}}{r! (3/2)_{r}} = \frac{1}{X} ∑_{r=0}^{∞} \frac{∂^{2r+1}}{∂u_{2}^{2r+1}} (\frac{−1}{\sqrt{u_{2}^{2} − 8u_{3}}} ) \frac{X^{2r+1}}{(2r + 1)!}

= \frac{1}{2X} (\frac{1}{\sqrt{(u_{2} − X)^{2} − 8u_{3}}} − \frac{1}{\sqrt{(u_{2} + X)^{2} − 8u_{3}}} ) ,  (161)

which is equivalent to formula (151) by a calculation similar to the one for s = 0.

This completes the proof of the part of Proposition 2 saying that y_{1} as an algebraic function. For the assertion that y itself is not algebraic in general, it suffices to prove the non-algebraicity of the specialization to σ_{1} = σ_{2} = σ_{3} = 0, i.e., to prove that the integral of the function (150) divided by x^{1/4} is not algebraic. We already know “morally” that it is not, because its first Taylor coefficients contain many different primes in the denominator, as was mentioned in the previous subsection, and this is also confirmed by asking a computer system like “Mathematica” to carry out the integral, obtaining as output the statement that there is no algebraic formula. A formal proof is as follows. We want to show that the integral w = ∫ u dt, where

u = \frac{1}{2\sqrt{P_{−}}} − \frac{1}{2\sqrt{P_{+}}} , with P_{±} = P_{±}(t) = 1 ± 6t^{2} + t^{4} ,

is not algebraic. By a theorem of Liouville, if w is algebraic it must belong to the function field ℚ(t, u) = ℚ(t) + ℚ(t)\sqrt{P_{+}} + ℚ(t)\sqrt{P_{−}} + ℚ(t)\sqrt{P_{+}P_{−}}. So we can assume that w = a + b\sqrt{P_{+}} + c\sqrt{P_{−}} + d\sqrt{P_{+}P_{−}} where a, b, c, d are rational functions of t. From w′ = u, we get 2b′P_{−} + bP′_{−} = 1, 2c′P_{+} + cP′_{+} = −1, and a = d = 0. By comparing the Laurent expansions around each t = α ∈ ℙ^{1}(ℂ) (distinguishing the cases P_{−}(α) = 0, P_{−}(α) ≠ 0 and α = ∞), we see that b has no poles on the complex projective line and hence is a constant, which is obviously a contradiction. This completes the proof of Proposition 2. □

Remark. We can recast the calculations just given for s = 0 and s = 1 into a slightly different and more suggestive form by using the differential equation L_{4}(G^{(4)}) = 0 directly rather than the expansion of G^{(4)} as an infinite linear combination of terms M_{4}^{r}(G^{(3)})σ_{4}^{r}. For instance, if s = 0 then we have

\frac{1}{2} L_{4} = \frac{1}{2} \frac{∂}{∂σ_{4}} + σ_{4} \frac{∂^{2}}{∂σ_{4}^{2}} − M_{4} = \frac{∂^{2}}{∂X^{2}} − M_{4} = (\frac{∂}{∂X} + \frac{∂}{∂u_{2}} )(\frac{∂}{∂X} − \frac{∂}{∂u_{2}} ) − \frac{∂^{2}}{∂u_{1}∂u_{3}} ,

so the unique function of u_{1}, u_{2}, u_{3} and X that is annihilated by L_{4}, even in X, and equal to a given function h(u_{2}, u_{3}) when X = 0 is simply \frac{1}{2}(h(u_{2} + X, u_{3}) + h(u_{2} − X, u_{3})), and we recover (160) without any need for using the infinite sum of differential operators occurring in eq. (115) and with almost no computation at all. Similarly, if s = 1 then we have

\frac{1}{2} L_{4} = X^{−1} \frac{∂^{2}}{∂X^{2}} X − M_{4} = X^{−1}((\frac{∂}{∂X} + \frac{∂}{∂u_{2}} )(\frac{∂}{∂X} − \frac{∂}{∂u_{2}} ) − \frac{∂^{2}}{∂u_{1}∂u_{3}} )X ,

so the unique function of u_{1}, u_{2}, u_{3} and X that is annihilated by L_{4}, even in X, and equal to ∂h(u_{2}, u_{3})/∂u_{2} when X = 0 is simply \frac{1}{2X} (h(u_{2} + X, u_{3}) − h(u_{2} − X, u_{3})), and we recover (161) easily.

This completes our discussion of the cases s = 0 and s = 1. To understand how the general case should go, we look at the next case s = 2 (d = 8). Here by direct calculation we find

y_{2}^{0} := \frac{1}{6}(E + 4)(E + 5)(E + 6)G^{(3)}(σ_{1}, σ_{2}, σ_{3}; 8)

= −2 \frac{3u_{2}(u_{2}^{2} − 8u_{3}) + u_{1}u_{2}(u_{2}^{2} + 12u_{3}) + 4u_{3}(u_{2}^{2} + 2u_{3})}{(u_{2}^{2} − 8u_{3})^{7/2}}

= \frac{1}{6} (3\frac{∂^{2}}{∂u_{2}∂u_{3}} + 2u_{1} \frac{∂^{3}}{∂u_{2}^{3}} )(\frac{1}{\sqrt{u_{2}^{2} − 8u_{3}}} ) + \frac{1}{12} \frac{∂^{3}}{∂u_{2}^{3}} (\frac{u_{2}}{\sqrt{u_{2}^{2} − 8u_{3}}} )

and hence

y_{2} = ∑_{r=0}^{∞} M_{4}^{r}(y_{2}^{0}) \frac{σ_{4}^{r}}{r! (5/2)_{r}} = ∑_{r=0}^{∞}(\frac{∂^{2r}}{∂u_{2}^{2r}} + r \frac{∂^{2r}}{∂u_{1}∂u_{2}^{2r−2}∂u_{3}} + O(\frac{∂^{2}}{∂u_{1}^{2}} ))(y_{2}^{0}) \frac{6(r + 1)X^{2r}}{(2r + 3)!}

= ∑_{r=0}^{∞}(r + 1)((2r + 3)\frac{∂^{2r+2}}{∂u_{2}^{2r+1}∂u_{3}} + 2u_{1} \frac{∂^{2r+3}}{∂u_{2}^{2r+3}} )(\frac{1}{\sqrt{u_{2}^{2} − 8u_{3}}} ) \frac{X^{2r}}{(2r + 3)!}

+ \frac{1}{2} ∑_{r=0}^{∞}(r + 1)\frac{∂^{2r+3}}{∂u_{2}^{2r+3}} (\frac{u_{2}}{\sqrt{u_{2}^{2} − 8u_{3}}} ) \frac{X^{2r}}{(2r + 3)!}

= \frac{2}{X} ∑_{r=0}^{∞} \frac{∂^{2r+1}}{∂u_{2}^{2r+1}} (\frac{1}{(u_{2}^{2} − 8u_{3})^{3/2}} )\frac{X^{2r+1}}{(2r + 1)!}

+ \frac{1}{4X} \frac{∂}{∂X} \frac{1}{X} ∑_{r=−1}^{∞} \frac{∂^{2r+3}}{∂u_{2}^{2r+3}} (\frac{u_{2} + 4u_{1}}{\sqrt{u_{2}^{2} − 8u_{3}}} ) \frac{X^{2r+3}}{(2r + 3)!}

= − \frac{1}{X} [\frac{1}{((u_{2} − X)^{2} − 8u_{3})^{3/2}} − \frac{1}{((u_{2} + X)^{2} − 8u_{3})^{3/2}} ]

− \frac{1}{8X} \frac{∂}{∂X} \frac{1}{X} [\frac{u_{2} − X + 4u_{1}}{\sqrt{(u_{2} − X)^{2} − 8u_{3}}} − \frac{u_{2} + X + 4u_{1}}{\sqrt{(u_{2} + X)^{2} − 8u_{3}}} ] .

This is an explicit algebraic function of σ_{1}, …, σ_{4} and we can check directly that it belongs to the function field generated over K by the function y_{0} or y_{1} or, more precisely, to the subspace of this field anti-invariant under its Galois automorphism over its subfield K(\sqrt{Δ_{2}}). After a good deal of computation we find that y_{2} is given with respect to the basis y_{0}, y_{1} of this space by

y_{2} = \frac{u_{2}Δ_{0}Δ_{1} − 4σ_{4}Δ_{1} − 2u_{2}Δ̃_{0}}{16σ_{4}Δ_{2}} y_{0} − \frac{Δ_{0}Δ_{1}^{2} − 8u_{2}σ_{4}(3u_{2}Δ_{0} − 4σ_{4}) + Δ̃_{0}Δ_{1}}{16σ_{4}Δ_{2}} y_{1} ,

with Δ̃_{0} = 4σ_{4}(Δ_{0} − u_{2} + 8), confirming Theorem 13 in this case.

Looking at the three cases treated so far, we see that all we needed in these cases was that the initial values y_{s}(0) = y_{s}(σ_{1}, σ_{2}, σ_{3}, 0) of the functions defined in (152) had the form

y_{0}(0) = h_{0,0} ,  y_{1}(0) = ∂_{2}h_{1,1} ,  y_{2}(0) = (\frac{3}{2}∂_{3} + u_{1}∂_{2}^{2})∂_{2}h_{2,1} + ∂_{2}^{3}h_{2,2}

for some functions h_{s,i} depending on u_{2} and u_{3} but not on u_{1}, where ∂_{p} = ∂/∂u_{p}. The same thing happens for the next case s = 3, the required decomposition this time having the form

y_{3}(0) = (\frac{15}{4}∂_{3}^{2} + 5u_{1}∂_{2}^{2}∂_{3} + u_{1}^{2}∂_{2}^{4})∂_{2}h_{3,1} + (\frac{5}{2}∂_{3} + u_{1}∂_{2}^{2})∂_{2}^{3}h_{3,2} + ∂_{2}^{5}h_{3,3}

for appropriately chosen functions h_{3,i} of u_{2} and u_{3}. Continuing to higher s, we find each time that y_{s}(0) has a decomposition

y_{s}(0) = ∑_{i=1}^{s} M_{s,i} h_{s,i}  (s ≥ 1)  (162)

where M_{s,i} (1 ≤ i ≤ s) are the differential operators

M_{s,i} = ∑_{j=i}^{s} \binom{s − \frac{1}{2}}{s − j} \frac{u_{1}^{j−i}}{(j − i)!} ∂_{3}^{s−j} ∂_{2}^{2j−1}  (163)

and h_{s,i} (1 ≤ i ≤ s) the functions given by

h_{s,i} = \frac{(−1)^{s}2^{s−i}(s − 1)!}{(2s − 1)! (i − 1)!} \frac{1}{\sqrt{Δ_{0}^{2} − 8σ_{3}}} (\frac{u_{2} − \sqrt{Δ_{0}^{2} − 8σ_{3}}}{2} )^{i−1} .  (164)

We verified (162) by computer up to s = 16, and will prove it for all s ∈ ℕ in the next two subsections. Here we show that it implies the algebraicity assertion in Theorem 13. This is a special case of the following more general statement.

Proposition 4. Let M_{s,i} (1 ≤ i ≤ s) be the operator defined in (163). Then the function

∑_{r=0}^{∞} \frac{σ_{4}^{r}}{r! (s + \frac{1}{2})_{r}} M_{4}^{r}(M_{s,i}h)

72 is algebraic for any algebraic function h of u_{2} and u_{3}.

Proof. Since h is independent of u_{1}, we have (using (156)) the identity

M_{4}^{r}(M_{s,i}h) = ( ∑_{l≥0} \binom{r}{l}∂_{2}^{2r−2l}∂_{1}^{l}∂_{3}^{l})(∑_{k=i}^{s} \binom{s − \frac{1}{2}}{s − k} \frac{u_{1}^{k−i}}{(k − i)!} ∂_{3}^{s−k} ∂_{2}^{2k−1}) h

= ∑_{j=i}^{s} (∑_{l=0}^{s−j} \binom{r}{l}\binom{s − \frac{1}{2}}{s − j − l}) \frac{u_{1}^{j−i}}{(j − i)!} ∂_{3}^{s−j} ∂_{2}^{2r+2j−1}h

= ∑_{j=i}^{s} \binom{r + s − \frac{1}{2}}{s − j} \frac{u_{1}^{j−i}}{(j − i)!} ∂_{3}^{s−j} ∂_{2}^{2r+2j−1}h .  (165)

It follows that

∑_{r=0}^{∞} \frac{σ_{4}^{r}}{r! (s + \frac{1}{2})_{r}} M_{4}^{r}(M_{s,i}h) = ∑_{j=i}^{s} \binom{s − \frac{1}{2}}{s − j}\frac{u_{1}^{j−i}}{(j − i)!} ∂_{3}^{s−j} (∑_{r=0}^{∞} \frac{σ_{4}^{r}}{r! (j + \frac{1}{2})_{r}} ∂_{2}^{2r+2j−1}h) ,

and this is algebraic because for each j ≥ 1 we have the identity

∑_{r=0}^{∞} \frac{σ_{4}^{r} ∂_{2}^{2r+2j−1}h}{r! (j + \frac{1}{2})_{r}} = (3/2)_{j−1} \frac{∂^{j−1}}{∂σ_{4}^{j−1}} ∑_{r=1−j}^{∞} \frac{(4σ_{4})^{r+j−1} ∂_{2}^{2r+2j−1}h}{(2r + 2j − 1)!}

= (3/2)_{j−1} \frac{∂^{j−1}}{∂σ_{4}^{j−1}} (\frac{h(u_{2} + X, u_{3}) − h(u_{2} − X, u_{3})}{2X})  (166)

with X = \sqrt{4σ_{4}} as in (159). □

Remark. There is an interesting relation between the formalism here and certain differential operators from smooth functions of one variable to functions of two variables introduced in [14]. These operators^{6} are defined by

D_{n}F (x, y) = \frac{(−1)^{n}}{n!} ∑_{i=0}^{n} \frac{(2n − i)!}{i! (n − i)!} \frac{(−1)^{i}F^{(i)}(x) − F^{(i)}(y)}{(x − y)^{2n−i+1}}  (n = 0, 1, 2, …) , (167)

the first three cases being

D_{0}F (x, y) = \frac{F (x) − F (y)}{x − y} ,
D_{1}F (x, y) = \frac{F′(x) + F′(y)}{(x − y)^{2}} − 2 \frac{F (x) − F (y)}{(x − y)^{3}} ,
D_{2}F (x, y) = \frac{1}{2} \frac{F′′(x) − F′′(y)}{(x − y)^{3}} − 3 \frac{F′(x) + F′(y)}{(x − y)^{4}} + 6 \frac{F (x) − F (y)}{(x − y)^{5}} .

Though D_{n}F (x, y) has an apparent pole of order 2n + 1 along the diagonal x = y, it is in fact smooth there. Other attractive properties of D_{n} proved in [14] are that it annihilates polynomials of degree ≤ 2n (as one can verify by hand in the above formulas for 0 ≤ n ≤ 2), that it is given on monomials x^{m} of degree ≥ 2n + 1 by

D_{n}(x ↦ x^{m})(x, y) = ∑_{r+s=m−1} \binom{r}{n}\binom{s}{n} x^{r−n}y^{s−n} ,

6with the slightly different notation and normalization (D_{n}F)(x, y) = n! (x − y)^{n+1}(D_{n}F)(x, y) in [14]

and that it satisfies the SL(2)-equivariance property D_{n}(F |_{−2n}g) = D_{n}(F)|_{−n−1,−n−1}g for all g ∈ SL(2, ℂ), where |_{k} (and similarly |_{k,k′}) denotes the usual “slash operator” (f|_{k}( \begin{matrix}a & b \\ c & d \end{matrix} ))(x) = (cx + d)^{−k}f(\frac{ax+b}{cx+d}). All of these follow from the integral representation ([14], Prop. 4(v))

D_{n}F (x, y) = \frac{1}{n!^{2} (x − y)^{2n+1}} ∫_{y}^{x}(z − y)^{n}(x − z)^{n}F^{(2n+1)}(z) dz

= \frac{1}{n!^{2}} ∫_{0}^{1} t^{n} (1 − t)^{n} F^{(2n+1)}(tx + (1 − t)y) dt .

We can also rewrite D_{n}F (x, y) in terms of the derivatives of F only at (x + y)/2, though at the expense of changing the finite sum (167) to an infinite one. To do this, we combine the integral representation just given with the usual Taylor expansion to find

(D_{n}F)(u + X, u − X) = \frac{1}{2^{2n+1}n!^{2}} ∫_{−1}^{1}(1 − t^{2})^{n} F^{(2n+1)}(u + tX) dt

= \frac{1}{2^{2n+1}n!^{2}} ∑_{r=0}^{∞}(∫_{−1}^{1} t^{2r}(1 − t^{2})^{n}dt) F^{(2r+2n+1)}(u) \frac{X^{2r}}{(2r)!} .

Computing the coefficient in parentheses by the beta integral and the duplication formula as

∫_{−1}^{1} t^{2r}(1 − t^{2})^{n}dt = ∫_{0}^{1} x^{r− \frac{1}{2}} (1 − x)^{n}dx = \frac{n! Γ(r + \frac{1}{2})}{Γ(n + r + \frac{3}{2})} = \frac{2^{2n+1}n!^{2}}{(2n + 1)!} \frac{(2r)!}{2^{2r}r! (n + \frac{3}{2})_{r}} ,

we find the following formula for the Taylor expansion of (D_{n}F)(u + X, u − X) :

(D_{n}F)(u + X, u − X) = \frac{1}{(2n + 1)!} ∑_{r=0}^{∞} F^{(2r+2n+1)}(u) \frac{(X/2)^{2r}}{r! (n + \frac{3}{2})_{r}} .  (168)

Together with (167) this gives another algebraic expression for the sum in eq. (166).

C. Properties of the generating function for fixed integral values of s. In the last subsection we reduced the proof of Theorem 13 to the proof of the experimentally discovered identity (162). In this subsection we will recast this identity in a nicer form and investigate some of its consequences, and will prove Theorem 14. The proof of the identity itself will follow from the generating function identities proved in §10.D.

Throughout this subsection we take as independent variables the variables u_{i} introduced in (155), and set ∂_{i} = ∂/∂u_{i}. We also set

Δ_{0} = (1 − σ_{1}/2)^{2} − σ_{2} = −u_{2} − 4u_{1} , Δ_{1} = Δ_{0}^{2} − 8σ_{3} = u_{2}^{2} − 8u_{3} ,

corresponding to the definitions in Propositions 1 and 2 at σ_{4} = 0. With these notations the quantity A_{±} occurring in (153) is given by A_{±} = \frac{Δ_{0}±\sqrt{Δ_{1}}}{2} .

The first observation is that for a function h that does not depend on u_{1} the operator (163) can be written in the more attractive form

M_{s,i}(h) = M_{4}^{s−1/2}(\frac{u_{1}^{s−i}}{(s − i)!} h) .  (169)

74 Here the fractional power M_{4}^{s−1/2} of the differential operator (156) is not defined as an operator, but makes sense on functions of u_{1}, u_{2}, u_{3} that are polynomials of degree ≤ s − 1 with respect to u_{1} if we use the binomial theorem to define

M_{4}^{s−1/2} = ∑_{l=0}^{s−1} \binom{s − 1/2}{l} ∂_{2}^{2s−2l−1}(∂_{1}∂_{3})^{l}  (170)

for such functions. The observation (169), whose proof is immediate from (170) and the definition of M_{s,i}, also clarifies the identity (166) appearing in the proof of Proposition 4 above, since the left-hand side of this identity can be written as M_{4}^{r+s−1/2}(u_{1}^{s−i}h/(s − i)!) and the fractional power M_{4}^{r+s−1/2} then expanded by the binomial theorem in the same way as was done in (170).

Substituting (169) and (164) into (162) and using the binomial theorem, we find that the expression appearing on the right-hand side of (162) is simply Y_{s}^{+}(0), where we define

Y_{s}^{±}(0) = Y_{s}^{±}(σ_{1}, σ_{2}, σ_{3}, 0) = −\frac{1}{(2s − 1)!} M_{4}^{s−1/2} (\frac{A_{±}^{s−1}}{\sqrt{Δ_{1}}} ) .  (171)

We will see in a moment that Y_{s}^{+}(0) = Y_{s}^{−}(0), corresponding to (but much easier than) the corresponding assertion for y_{s}^{±}(0) in Theorem 14. If, in analogy with (154), we also define

Y_{s}^{±} = Y_{s}^{±}(σ_{1}, σ_{2}, σ_{3}, σ_{4}) = ∑_{r=0}^{∞} M_{4}^{r}(Y_{s}^{±}(0)) \frac{σ_{4}^{r}}{r! (s + \frac{1}{2})_{r}} ,  (172)

then the identities to be proved are

y_{s}^{+}(σ_{4}) = y_{s}^{−}(σ_{4}) = Y_{s}^{+}(σ_{4}) = Y_{s}^{−}(σ_{4}) ,  (173)

where we have suppressed the dependence on σ_{1}, σ_{2} and σ_{3}. It clearly suffices to prove this for σ_{4} = 0, since each of the four functions in (173) is obtained from its value at σ_{4} = 0 by applying the operator ∑_{r≥0}(σ_{4}^{r}/r!(s + \frac{1}{2})_{r})M_{4}^{r}.

Before proceeding we write the formula for Y_{s}^{±}(σ_{4}) in a more explicit form. From the definition (170) (with s replaced by r + s) and the formula ∂_{1}(A^{±}) = −2 we get

M_{4}^{r+s− \frac{1}{2}} (\frac{A_{±}^{s−1}}{\sqrt{Δ_{1}}} ) = ∑_{l=0}^{s−1} \binom{r + s − \frac{1}{2}}{l} ∂_{2}^{2r+2s−2l−1}∂_{1}^{l}∂_{3}^{l} (\frac{A_{±}^{s−1}}{\sqrt{Δ_{1}}} )

= ∑_{l+n=s−1} \frac{(−1/2)^{l}(2s − 1)! (s + \frac{1}{2})_{r}}{l! (2n + 1)! (n + \frac{3}{2})_{r}} ∂_{3}^{l}∂_{2}^{2r+2n+1}(\frac{A_{±}^{n}}{\sqrt{Δ_{1}}} )

and hence

Y_{s}^{±}(σ_{1}, σ_{2}, σ_{3}, σ_{4}) = − ∑_{l+n=s−1} \frac{(−1/2)^{l}}{l!} ∂_{3}^{l} [\frac{1}{(2n + 1)!} ∑_{r=0}^{∞} \frac{σ_{4}^{r}}{r! (n + \frac{3}{2})_{r}} ∂_{2}^{2r+2n+1}(\frac{A_{±}^{n}}{\sqrt{Δ_{1}}} )] .

We can now use (168) to write the inner sum as D_{n}(A_{±}^{n}/\sqrt{Δ_{1}})(u_{2} + X, u_{2} − X), giving an explicit expression for Y_{s}^{±} as an algebraic function of its arguments and showing that it belongs to the space described in Theorem 13. We also see immediately that Y_{s}^{+} = Y_{s}^{−}, because (A_{+}^{n} − A_{−}^{n})/\sqrt{Δ_{1}} is a polynomial of degree at most n − 1 in u_{2} and hence is annihilated by ∂_{2}^{2r+2n+1} (or even by ∂_{2}^{n}). Finally, we can use simple binomial coefficient identities to rewrite the formula in a slightly different form, which will be used in §10D:

Y_{s}^{±}(σ_{1}, σ_{2}, σ_{3}, σ_{4}) = − ∑_{l+n=s−1} \frac{(−1/2)^{l}}{l!} ∂_{3}^{l} [∑_{r=0}^{∞} \frac{\binom{n+r}{r} (4σ_{4})^{r}}{(2r + 2n + 1)!} ∂_{2}^{2r+2n+1}(\frac{A_{±}^{n}}{\sqrt{Δ_{1}}} )] . (174)

We end the subsection by giving the proof of Theorem 14. As already noted, it suffices to show that y_{s}^{+}(0) = y_{s}^{−}(0). We observe first that the form of the functions y_{s}^{±}(0) can be simplified:

y_{s}^{±}(0) = \binom{E + 3s}{2s − 1} \frac{A_{±}^{−s}}{\sqrt{Δ_{1}}} = \binom{E + 3s}{2s − 1} (2σ_{3})^{−s} \frac{A_{∓}^{s}}{\sqrt{Δ_{1}}} = \frac{1}{(2σ_{3})^{s}} \binom{E}{2s − 1} \frac{A_{∓}^{s}}{\sqrt{Δ_{1}}} , (175)

where we have used A_{+}A_{−} = 2σ_{3} and E(σ_{3}) = 3σ_{3}. The proof of the desired identity y_{s}^{+}(0) = y_{s}^{−}(0) now depends on the following proposition.

Proposition 5. Define polynomials B_{k}^{(s)} = B_{k}^{(s)}(σ_{1}, σ_{2}, σ_{3}) (k ≥ 0) by the generating function

B^{(s)}(x) = ∑_{k=0}^{∞} B_{k}^{(s)}x^{k} = \frac{1}{\sqrt{Δ̃_{0}^{2} + 8σ_{3}x^{3}(1 − x)}}(\frac{Δ̃_{0} + \sqrt{Δ̃_{0}^{2} + 8σ_{3}x^{3}(1 − x)}}{2})^{s} , (176)

where Δ̃_{0} = Δ̃_{0}(x) = 1 + 2u_{1}x + Δ_{0}x^{2}. Then we have the identities

(E − k)B_{k}^{(s)} = (2s − k − 1)B_{k−1}^{(s)}  (177)

for all k ≥ 0 (with B_{−1}^{(s)} := 0) and

B_{2s−2}^{(s)} = \frac{A_{+}^{s} − A_{−}^{s}}{\sqrt{Δ_{1}}}  (178)

Since the definition (176), discovered only after considerable experimentation, is not very enlightening, we give a few examples of the functions B_{k}^{(s)} and properties (177) and (178). From B^{(s)}(x) ≡ R^{s−1} (mod x^{3}) we see that the first three values of B_{k}^{(s)} are given by

B_{0}^{(s)} = 1 ,  B_{1}^{(s)} = 2(s − 1)u_{1} ,  B_{2}^{(s)} = 2(s − 1)(s − 2)u_{1}^{2} + (s − 1)Δ_{0} .

From this, using E(u_{1}) = u_{1} + 1 and E(Δ_{0}) = 2Δ_{0} + 2u_{1}, we find

E(B_{0}^{(s)}) = E(1) = 0 ,
(E − 1)(B_{1}^{(s)}) = 2(s − 1)(E − 1)(u_{1}) = (2s − 2)B_{0}^{(s)},
(E − 2)(B_{2}^{(s)}) = 2(s − 1)(s − 2)(2u_{1}) + (s − 1)(2u_{1}) = (2s − 3)B_{1}^{(s)} ,

verifying (177) for these three values, while for (178) we have

B_{0} = 1 = \frac{A_{+} − A_{−}}{\sqrt{Δ_{1}}} , B_{2}^{(2)} = Δ_{0} = \frac{A_{+}^{2} − A_{−}^{2}}{\sqrt{Δ_{1}}} .

The proof that Proposition 5 implies Theorem 14 is almost immediate. Equation (177) and induction on n give \binom{E−2s+1+n}{n}B_{2s−2}^{(s)} = B_{2s−2−n}^{(s)} for all n ≥ 0. In particular, \binom{E}{2s−1}B_{2s−2}^{(s)} vanishes, and in view of (178) and (175) this proves the theorem. It remains to prove the proposition.

Proof of (177): We have to show that

E(B^{(s)}) = x[(1 − x)\frac{∂B^{(s)}}{∂x} + (2s − 2)B^{(s)}] .

Define Ẽ = \frac{1}{x}E − (1 − x)\frac{∂}{∂x}, which is still a derivation. Also set Z = σ_{3}x^{3}(1 − x), so that

B^{(s)} = \frac{1}{\sqrt{Δ̃_{0}^{2} + 8Z}} (\frac{Δ̃_{0} + \sqrt{Δ̃_{0}^{2} + 8Z}}{2} )^{s} ,

Then

Ẽ(Z) = \frac{3Z}{x} − (1 − x)σ_{3}(3x^{2} − 4x^{3}) = 4Z
Ẽ(Δ̃_{0}) = 2(u_{1} + 1) + 2(u_{1} + Δ_{0})x − (1 − x)(2u_{1} + 2Δ_{0}x) = 2Δ̃_{0} ,

In other words, Δ̃_{0} and Z are of degree 2 and 4 with respect to the modified Euler operator Ẽ, so Ẽ(B^{(s)}) = 2(s − 1)B^{(s)} as desired.

Proof of (178): Since Δ̃_{0} − \sqrt{Δ̃_{0}^{2} + 8Z} = O(x^{3}), we can replace B^{(s)} by

B̃^{(s)} = \frac{1}{\sqrt{Δ̃_{0}^{2} + 8Z}} [(\frac{Δ̃_{0} + \sqrt{Δ̃_{0}^{2} + 8Z}}{2} )^{s} + (\frac{Δ̃_{0} − \sqrt{Δ̃_{0}^{2} + 8Z}}{2} )^{s}]

without changing the value of the coefficient of x^{2s−2}. But B̃^{(s)} is a polynomial, given explicitly by

2^{s−1}B̃^{(s)} = ∑_{0≤2j≤s−1} \binom{s}{2j + 1}Δ̃_{0}^{s−2j−1}(Δ̃_{0}^{2} + 8Z)^{j}

= ∑_{0≤2j≤s−1} \binom{s}{2j + 1}(Δ_{0}x^{2} + ⋯)^{s−2j−1}((Δ_{0}^{2} − 8σ_{3})x^{4} + ⋯)^{j}

= x^{2s−2} \frac{(Δ_{0} + \sqrt{Δ_{1}})^{s} − (Δ_{0} − \sqrt{Δ_{1}})^{s}}{2 \sqrt{Δ_{1}}} + ⋯,

where “⋯ ” denotes lower order terms as x → ∞. The result follows.

Remark. If we replace the expressions Δ_{0} = (1 − σ_{1}/2)^{2} − σ_{2} and Δ_{1} = Δ_{0}^{2} − 8σ_{3} by the corresponding homogeneous forms

Δ_{0}(u, v) = (u − σ_{1}v/2)^{2} − σ_{2}v^{2} , Δ_{1}(u, v) = Δ_{0}(u, v)^{2} − 8σ_{3}uv^{3}

of degrees 2 and 4, respectively, in u and v, then Δ̃_{0} = Δ_{0}(1−x, x), Δ̃_{0}^{2} +8Z = Δ_{1}(1−x, x), and the strange-looking “modified Euler operator” Ẽ used in the proof above is simply the ordinary Euler operator u ∂_{u} + v ∂_{v} with respect to the homogeneous variables u and v.

D. Generating function with respect to s. Since we have shown that both y_{s}^{±} and Y_{s}^{±} are independent of the choice of sign, we can write them simply as y_{s} and Y_{s}, respectively, and to obtain (173) it will suffice to prove that Y_{s}(0) = y_{s}(0). We will not do this directly, but rather by proving (for σ_{4} = 0) the equality of the two generating functions

Y(T) = Y(σ_{1}, σ_{2}, σ_{3}, σ_{4}; T) = ∑_{s=1}^{∞} Y_{s}(σ_{1}, σ_{2}, σ_{3}, σ_{4}) T^{s−1} ,  (179)

y(T) = y(σ_{1}, σ_{2}, σ_{3}, σ_{4}; T) = ∑_{s=1}^{∞} y_{s}(σ_{1}, σ_{2}, σ_{3}, σ_{4}) T^{s−1}  (180)

with respect to the index s. This approach gives the additional dividend that the generating function y(T) turns out itself to be an algebraic function. This of course implies the algebraicity of each of its Taylor coefficients y_{s}, but it does not make superfluous the algebraic calculations of the last two subsections because, rather surprisingly, the degree drops under differentiation in T and specialization to T = 0 : the function y(T) will turn out to be algebraic of degree 6 over ℚ(σ_{1}, …, σ_{4}, T), whereas Theorem 13 says that each individual y_{s}(T) satisfies an algebraic equation of degree only 4.

Equation (174) simplifies considerably when we substitute it into the generating series (179), and we obtain:

Y(T) = ∑_{l=0}^{∞} \frac{(−T/2)^{l}}{l!} ∂_{3}^{l} [ ∑_{m=0}^{∞} \frac{∂_{2}^{2m+1}}{(2m + 1)!} ( ∑_{n+r=m} \binom{m}{n} \frac{(4σ_{4})^{r}(T A_{±})^{n}}{\sqrt{Δ_{1}}} )]

= ∑_{m=0}^{∞} \frac{∂_{2}^{2m+1}}{(2m + 1)!} [∑_{l=0}^{∞} \frac{(−T/2)^{l}}{l!} ∂_{3}^{l} (\frac{(4σ_{4} + T A_{±})^{m}}{\sqrt{Δ_{1}}} )]

= ∑_{m=0}^{∞} \frac{∂_{2}^{2m+1}}{(2m + 1)!} [\frac{1}{\sqrt{Δ_{1} + 4T}} (4σ_{4} + T \frac{Δ_{0} ± \sqrt{Δ_{1} + 4T}}{2} )^{m}] ,  (181)

where in the last line we have used the Taylor expansion and the fact that changing u_{3} to u_{3} − T/2 leaves Δ_{0} unchanged but changes Δ_{1} to Δ_{1} + 4T.

This expression can be further simplified by using Lagrange’s theorem. Recall that this theorem is an identity generalizing Taylor’s theorem which, in one of its forms, states that

∑_{n=0}^{∞} \frac{1}{n!} \frac{d^{n}}{dx^{n}} (f(x) b(x)^{n}) = \frac{f(z)}{1 − b′(x)}|_{z = x+b(z)}  (182)

whenever the series makes sense (e.g., when everything is complex and absolutely convergent, or when the variables are power series in x with b(x) = O(x^{2}), or when b is a power series in some other variable with positive valuation). The corresponding “odd Lagrange theorem,” obtained by replacing b(x) and f(x) by ±\sqrt{B(x)} and F (x)/\sqrt{B(x)} and taking the difference of the two equations obtained, says that

∑_{m=0}^{∞} \frac{1}{(2m + 1)!} \frac{d^{2m+1}}{dx^{2m+1}} (F (x) B(x)^{m}) = ∑_{±} \frac{F (z)}{2(z − x) − B′(z)}|_{z = x±\sqrt{B(z)}} .  (183)

Applying this (with x and z −x replaced by u_{2} and w) to (181), we find after a short calculation that the two formulas (181) can be written in the form

Y(T) = \frac{T}{P′(w_{1})} + \frac{T}{P′(w_{2})} = − \frac{T}{P′(w_{3})} − \frac{T}{P′(w_{4})} ,  (184)

78 where w_{1} and w_{2} (resp. w_{3} and w_{4}) are the solutions of

w_{1} = \sqrt{4σ_{4} + T A_{+}(u_{1}, u_{2} + w_{1}, u_{3} − T/2)} , w_{2} = −\sqrt{4σ_{4} + T A_{+}(u_{1}, u_{2} + w_{2}, u_{3} − T/2)}

(resp. of the corresponding equations with A_{+} replaced by A_{−}) and where

P (w) = (w^{2} − 4σ_{4} − T Δ_{0}(u_{1}, u_{2} + w, u_{3} − T/2)/2)^{2} − T^{2}Δ_{1}(u_{1}, u_{2} + w, u_{3} − T/2)/4
= w^{4} + T w^{3} − (8σ_{4} + Δ_{0}T)w^{2}
− (4σ_{4}T − 2u_{1}T^{2})w + (16σ_{4}^{2} + 4Δ_{0}σ_{4}T + 2σ_{3}T^{2} − T^{3})  (185)

is the monic quartic polynomial having w_{1}, …, w_{4} as its roots. The equality of the two expressions in (184), which appeared there because we had shown computationally that Y_{s}^{+} = Y_{s}^{−} for each s, now has a much more natural proof via the residue theorem, since

∑_{i=1}^{4} \frac{1}{P′(w_{i})} = ∑_{α∈ℙ^{1}(ℂ)} Res_{w=α}(\frac{dw}{P (w)}) = 0 .  (186)

We now turn to the second generating function (180), which we will evaluate by using the odd Lagrange theorem again in combination with the formula (175). To achieve this, we introduce a “dummy” variable of homogeneity x by setting σ_{p} = A_{p}x^{P} (1 ≤ p ≤ 4), so that

Δ_{0} = (1 − A_{1}x/2)^{2} − A_{2}x^{2} , Δ_{1} = Δ_{0}^{2} − 8A_{3}x^{3} , A_{±} = (Δ_{0} ± \sqrt{Δ_{1}})/2

become functions of x and the Euler operator E becomes simply x d/dx. Then (175) gives

y(0, T) = ∑_{s=1}^{∞} \frac{T^{s−1}}{(2σ_{3})^{s}} \binom{E}{2s − 1} \frac{A_{±}^{s}}{\sqrt{Δ_{1}}} = \frac{1}{2A_{3}x^{2}} ∑_{m=0}^{∞} \frac{(T/2A_{3}x)^{m}}{(2m + 1)!} \frac{d^{2m+1}}{dx^{2m+1}} (\frac{A_{±}^{m+1}}{\sqrt{Δ_{1}}} ) ,

and applying (183) we find

y(0, T) = ∑_{i=1}^{2} \frac{A_{+}(z_{i})/\sqrt{Δ_{1}(z_{i})}}{2(z_{i} − x) − A′_{+}(z_{i})} = − ∑_{i=3}^{4} \frac{A_{−}(z_{i})/\sqrt{Δ_{1}(z_{i})}}{2(z_{i} − x) − A′_{−}(z_{i})}  (187)

where z_{1} and z_{2} (resp. z_{3} and z_{4}) are the solutions of the equations

z_{1} = x + \sqrt{\frac{T A_{+}(z_{1})}{2A_{3}x}} ,  z_{2} = x − \sqrt{\frac{T A_{+}(z_{2})}{2A_{3}x}}  (188)

(resp. of the same equations with A_{+} replaced by A_{−}). Now substituting back A_{p} = σ_{p}/x^{p} into the equations (188), we find that they are homogeneous (of degree 1) in z and x, so that only the ratio z/x matters, and making the change of variables w = T z/(z − X) we find that the four numbers z_{i} correspond exactly to the four numbers w_{i} defined above and that the right-hand side of (187) coincides exactly with that of (184) with σ_{4} specialized to 0. This finally completes the proof of (173) and hence of Theorem 13,

We end by formulating a theorem describing the algebraic nature of the generating function(s) we have just computed.

Theorem 15. The square of the function y(σ_{1}, …, σ_{4}; T) defined in (180) is algebraic of degree 3 over ℚ(σ_{1}, …, σ_{4}, T).

Proof. This follows almost immediately from (184). Let N = K(w_{1}, …, w_{4}) denote the splitting field of the polynomial P (w) defined in (185) over the field K = ℚ(σ_{1}, …, σ_{4}, T). The fact that the trace of h_{i} := T/P′(w_{i}) vanishes implies that the set {(h_{1} + h_{2})^{2}, (h_{1} + h_{3})^{2}, (h_{1} + h_{4})^{2}} is invariant under the Galois group of N over K, so the polynomial Q(t) = ∏_{i=2}^{4}(t − (h_{1} + h_{i})^{2}) has coefficients in K, and equation (184) says that Y^{2} is a root of this polynomial. □

We can get an explicit expression for Q(t) using Ferrari’s formula. If w is a root of the polynomial P (w) defined in (185), then h = T/P′(w) belongs to the field generated by w and hence also satisfies a fourth degree equation. This equation has a vanishing cubic term because of eq. (186), and by direct computation we find that it is given explicitly by

a_{4}h^{4} + 0h^{3} + a_{2}h^{2} + a_{1}h + 1 = 0 ,

where

a_{1} = T (T − 4u_{2}) ,
a_{2} = 6(u_{1} − 1)T^{3} + (72u_{1}^{2} + 4Δ_{0}(7u_{1} − 4) + \frac{1}{2}Δ_{0}^{2} − \frac{3}{2}Δ_{1} − 12σ_{4}) T^{2}
− 4(Δ_{0}Δ_{1} + 4(8 + 4u_{1} + 3Δ_{0})σ_{4}) T − 32σ_{4}(Δ_{1} + 4σ_{4}) ,
a_{4} = \frac{disc(P)}{T^{4}} = −27T^{6} + ⋯ + 256σ_{4}^{2} ((Δ_{1} + 4σ_{4})^{2} − 16u_{2}^{2}σ_{4}) .

Ferrari’s formula tells us that the cubic polynomial having as its roots the three numbers

(h_{1} + h_{2})^{2} = (h_{3} + h_{4})^{2}, (h_{1} + h_{3})^{2} = (h_{2} + h_{4})^{2}, (h_{1} + h_{4})^{2} = (h_{2} + h_{3})^{2}

is given by

Q(t) = a_{4}^{2} t^{3} + 2a_{2}a_{4} t^{2} + (a_{2}^{2} − 4a_{4}) t − a_{1}^{2} .

Then Q(y^{2}) = 0 is the sextic equation satisfied by our generating function y = y(σ_{1}, …, σ_{4}, T).

E. Odd values of d. As already mentioned in A. above, for odd values of d we were not able to find any closed formula for the function G^{(4)}, either as an algebraic function or an integral of an algebraic function or as the solution of a linear differential equation with polynomial coefficients. On the other hand, in another respect these functions were actually better behaved than they were for even d. Namely, unlike the case of even d, where the coefficients of the power series for d > 4 had infinitely many prime numbers occurring in their denominators and where each successive value of d required more differentiation of G^{(4)} in order to achieve integrality, here the power series are apparently always integral after rescaling x by a fixed factor. For instance, for d = 5 the same specialization G^{(4)}(0, 0, 0, x/4) as we used in §10.A has a Taylor series beginning

y = 1 + \frac{155}{16} x + \frac{194985}{1024} x^{2} + \frac{74509435}{16384} x^{3} + \frac{499522579465}{4194304} x^{4} + ⋯,  (189)

and within the limits of our calculation belongs to ℤ[[x/64]]. Moreover, this integrality holds not only for this particular specialization, but (experimentally) for the full generating series G^{(4)}(σ_{1}, σ_{2}, σ_{3}, σ_{4}; d = 5), which begins

1+ \frac{3σ_{1}}{2} + \frac{3σ_{1}^{2} + 3σ_{2}}{2} + \frac{5σ_{1}^{3} + 15σ_{1}σ_{2} + 20σ_{3}}{4} + \frac{15σ_{1}^{4} + 90σ_{1}^{2}σ_{2} + 280σ_{1}σ_{3} + 30σ_{2}^{2} + 620σ_{4}}{16} + · · · .

and in which the coefficient of degree k (up to k = 194, the limit of our computation) is always 2^{−k} times a polynomial in the σ_{p} with integral coefficients. (The specialized series (189) has an even stronger integrality property: up to a power of 2, the coefficient of x^{m}, corresponding to degree 4m, seems always to be divisible by 4m + 1. The corresponding property for the full series, which would say that the terms of total degree k are divisible by k +1, is not always true. However, we do find that all the terms of total degree p − 1 or p − 2 are divisible by p if p is an odd prime number, as one can see in the case p = 5 in the expansion given above.) Similarly, for d = 7, 9, 11 and 13 we find that the terms of degree k in G^{(4)}(σ_{1}, σ_{2}, σ_{3}, σ_{4}; d) are integral after multiplication by 2^{k}D_{d} in the range tested, where D_{7} = 1, D_{9} = 3, D_{11} = 5 and D_{13} = 35. Based on these experimental results, we formulate:

Conjecture. After rescaling σ_{p} ↦ 2^{p}σ_{p} (1 ≤ p ≤ 4), the generating series G^{(4)}(σ_{1}, σ_{2}, σ_{3}, σ_{4}; d) has a bounded denominator for every odd value of d ≥ 5.

In view of this conjecture, we can wonder whether the generating series for odd d are perhaps algebraic after all, even though we were not able to recognize even the special case (189) as an algebraic function. The following argument strongly suggests that they are not algebraic, and furthermore that—unlike what we saw in the case of even d—this remains true even after taking their images under any polynomial in the Euler operator. (This is a little surprising since the polynomial in E that we needed for d even, namely \binom{E+3s}{2s−1}, makes sense also for d ≥ 5 odd, since 2s − 1 is then still a non-negative integer.) Denote by a_{n} the coefficient of x^{n} in the power series (189). Since both of the special functions (149) and (150), corresponding to G^{(4)} and its derivative under E + 3 for d = 4 and d = 6, respectively, had singularities at the root x = (1 + \sqrt{2})^{−4} of 1 − 34x + x^{2} = 0, we expect these coefficients to grow asymptotically like C^{n}, where C = (1 + \sqrt{2})^{4} = 17 + 12\sqrt{2}. In fact, applying the numerical extrapolation method that is described in [17] and in §5.1 of [6], we find the conjectural asymptotic expansion

a_{n} ∼^{?} \frac{A C^{n}}{n} (1 − \frac{b_{1}}{n} + \frac{b_{2}}{n^{2}} − ⋯)  (190)

with coefficients A, b_{1}, b_{2}, … given numerically to high precision by

A ≈ 0.3900174223865606939417147 ,
b_{1} ≈ 0.35826456543968 ,  b_{2} ≈ 0.1163559071350 ,  ….

We can recognize these numbers as

A =^{?} \frac{(1 + \sqrt{2})^{2}}{2^{9/4} π} ,  (191)

and

b_{1} =^{?} 5 \frac{6 − \sqrt{2}}{2^{6}} ,  b_{2} =^{?} 5 \frac{183 − 62\sqrt{2}}{2^{12}} ,
b_{3} =^{?} 5 \frac{5802 − 2773\sqrt{2}}{2^{18}} ,  b_{4} =^{?} 5 \frac{389055 − 211804\sqrt{2}}{2^{25}} ,
b_{5} =^{?} 5 \frac{13542450 − 7562711\sqrt{2}}{2^{31}} ,  b_{6} =^{?} 5 \frac{469116531 − 283263874\sqrt{2}}{2^{37}} ,

where we can assert with a very high degree of confidence that these numerically guessed values are in fact correct. (In the case of A, for example, which is the most important value, the extrapolation method was first applied using only half of the known coefficients a_{n} to give a 15-digit prediction; this was then recognized as the number given in (191); and then a new calculation using all of the known coefficients gave a 25-digit value for A whose first 24 digits agreed with the prediction.) Formula (190) says that the function y has a logarithmic singularity at x = C^{−1}, which is incompatible with its being an algebraic function but would not preclude its being the integral of an algebraic function, but (190) and (191) together, if they are true, imply that P (E) y cannot be algebraic for any polyomial P (E) in the Euler operator E with integral coefficients, since the leading coefficient in the Laurent expansion of this function around x = 1/C would equal π times an algebraic number, and a function that is algebraic over ℂ(x) and belongs to ℚ[[x]] has to be algebraic over ℚ(x) and hence must have algebraic Laurent coefficients in its expansion at any algebraic point.

F. Final remarks. We do not know how to generalize to higher values of n the results that we found in this section for n = 4. In particular, it is hard to see how to generalize the change of variables (155) from n = 3 to larger n, and even harder to see how the change of variables (155) and (159) can be generalized from n = 4 to larger n. However, we did find a change of variables that makes the differential operators M_{n} simpler for all n, and that seems worth mentioning. Define new variables τ_{p} (p = 1, 2, …) by the generating function equality

\sqrt{1 + σ_{1}x + σ_{2}x^{2} + σ_{3}x^{3} + ⋯} = 1 + τ_{1}x + τ_{2}x^{2} + τ_{3}x^{3} + ⋯,

so that the first τ_{p} are given by

τ_{1} = \frac{1}{2}σ_{1} ,
τ_{2} = \frac{1}{2}σ_{2} − \frac{1}{8}σ_{1}^{2} ,
τ_{3} = \frac{1}{2}σ_{3} − \frac{1}{4}σ_{1}σ_{2} + \frac{1}{16}σ_{1}^{3} ,
τ_{4} = \frac{1}{2}σ_{4} − \frac{1}{4}σ_{1}σ_{3} − \frac{1}{8}σ_{2}^{2} + \frac{3}{16}σ_{1}^{2}σ_{2} − \frac{5}{128}σ_{1}^{4} .

Then an easy calculation shows that

M_{n} = 4 ∑_{i+j=n} \frac{∂^{2}}{∂τ_{i}∂τ_{j}} ,  (192)

which is simpler than the original definition (113) in two respects:

• there are only O(n) rather than O(n^{2}) terms, and

• the terms belong to the commutative algebra ℚ(∂/∂τ_{p})_{p≥1} rather than to the non-

commutative algebra ℚ(σ_{p}, ∂/∂σ_{p})_{p≥1}. Unfortunately, we have not been able to see how to use these new variables to make any of the generating function calculations simpler.

Tables of higher spherical polynomials

Table 1. Spherical polynomials for n = 3

We give a table of the polynomials P_{ν ν ν,d}(T) for all ννν = (ν_{1}, ν_{2}, ν_{3}) with |ννν| ≤ 5, extending the one given in the special case d = 4 after Proposition 1 of §8. As there, we give only one representative for each 𝔖_{3}-orbit of indices ννν. As usual, we write s for (d − 4)/2 and (x)_{r} for the ascending Pochhammer symbol x(x + 1) ⋯ (x + r − 1).

|ννν| = 0 :

P_{000} = 1

|ννν| = 1 :

P_{100} = (s + 1) r_{1}

|ννν| = 2 :

P_{200} = \frac{(s + 1)_{2}}{2} r_{1}^{2} − (s + 1) m_{2}m_{3}

P_{110} = (s + 1)_{2} r_{1}r_{2} − (s + 1) m_{3}r_{3}

|ννν| = 3 :

P_{300} = \frac{(s + 1)_{3}}{6} r_{1}^{3} − (s + 1)_{2} m_{2}m_{3}r_{1}

P_{210} = \frac{(s + 1)_{3}}{2} r_{1}^{2}r_{2} − (s + 1)_{2} m_{3}(r_{1}r_{3} + m_{2}r_{2})

P_{111} = (s + 2)^{3} r_{1}r_{2}r_{3} − (s + 2)^{2} (m_{1}r_{1}^{2} + m_{2}r_{2}^{2} + m_{3}r_{3}^{2}) + 4(s + 2) m_{1}m_{2}m_{3}

|ννν| = 4 :

P_{400} = \frac{(s + 1)_{4}}{24} r_{1}^{4} − \frac{(s + 1)_{3}}{2} m_{2}m_{3}r_{1}^{2} + \frac{(s + 1)_{2}}{2} m_{2}^{2}m_{3}^{2}

P_{310} = \frac{(s + 1)_{4}}{6} r_{1}^{3}r_{2} − \frac{(s + 1)_{3}}{2} m_{3}r_{1}(r_{1}r_{3} + 2m_{2}r_{2}) + (s + 1)_{2} m_{2}m_{3}^{2}r_{3}

P_{220} = \frac{(s + 1)_{4}}{4} r_{1}^{2}r_{2}^{2} − \frac{(s + 1)_{3}}{2} m_{3}(2r_{1}r_{2}r_{3} + m_{1}r_{1}^{2} + m_{2}r_{2}^{2})

+ \frac{(s + 1)_{2}}{2} m_{3}^{2}(r_{3}^{2} + 2m_{1}m_{2})

P_{211} = \frac{(s + 2)^{2}(s + 3)^{2}}{2} r_{1}^{2}r_{2}r_{3} − \frac{(s + 2)(s + 3)^{2}}{2} m_{1}r_{1}^{3}

− (s + 2)^{2}(s + 3) r_{1}(m_{2}r_{2}^{2} + m_{3}r_{3}^{2}) − (s + 1)(s + 2)^{2} m_{2}m_{3}r_{2}r_{3} + (s + 2)(5s + 13) m_{1}m_{2}m_{3}r_{1}

|ννν| = 5 :

P_{500} = \frac{(s + 1)_{5}}{120} r_{1}^{5} − \frac{(s + 1)_{4}}{6} m_{2}m_{3}r_{1}^{3} + \frac{(s + 1)_{3}}{2} m_{2}^{2}m_{3}^{2}r_{1}

P_{410} = \frac{(s + 1)_{5}}{24} r_{1}^{4}r_{2} − \frac{(s + 1)_{4}}{6} m_{3}r_{1}^{2}(r_{1}r_{3} + 3m_{2}r_{2}) + \frac{(s + 1)_{3}}{2} m_{2}m_{3}^{2}(2r_{1}r_{3} + m_{2}r_{2})

P_{320} = \frac{(s + 1)_{5}}{12} r_{1}^{3}r_{2}^{2} − \frac{(s + 1)_{4}}{6} m_{3}r_{1}(3r_{1}r_{2}r_{3} + m_{1}r_{1}^{2} + 3m_{2}r_{2}^{2})

+ \frac{(s + 1)_{3}}{2} m_{3}^{2} (r_{1}r_{3}^{2} + 2m_{2}r_{2}r_{3} + 2m_{1}m_{2}r_{1})

P_{311} = \frac{(s + 2)_{3}}{6} ((s + 2)(s + 4) r_{1}^{3}r_{2}r_{3} − (s + 4) m_{1}r_{1}^{4} − 3(s + 2) r_{1}^{2}(m_{2}r_{2}^{2} + m_{3}r_{3}^{2}))

− (s + 2)_{2} ((s + 2)^{2} m_{2}m_{3}r_{1}r_{2}r_{3} − (3s + 10)m_{1}m_{2}m_{3}r_{1}^{2}

− (s + 2)m_{2}m_{3}(m_{2}r_{2}^{2} + m_{3}r_{3}^{2}) + 4 m_{1}m_{2}^{2}m_{3}^{2})

P_{221} = \frac{(s + 2)(s + 3)^{2}}{2} (\frac{(s + 3)(s + 4)}{2} r_{1}^{2}r_{2}^{2}r_{3} − (s + 4) r_{1}r_{2}(m_{1}r_{1}^{2} + m_{2}r_{2}^{2})

− 2(s + 3) m_{3}r_{1}r_{2}r_{3}^{2} − s m_{3}r_{3}(m_{1}r_{1}^{2} + m_{2}r_{2}^{2}) + m_{3}^{2}r_{3}^{3}

+ 12 m_{1}m_{2}m_{3}r_{1}r_{2}) + (s + 2)(s + 3)(s − 3) m_{1}m_{2}m_{3}^{2}r_{3}

|ννν| = 6 :

P_{600} = \frac{(s + 1)_{6}}{720} r_{1}^{6} − \frac{(s + 1)_{5}}{24} m_{2}m_{3}r_{1}^{4} + \frac{(s + 1)_{4}}{4} m_{2}^{2}m_{3}^{2}r_{1}^{2} − \frac{(s + 1)_{3}}{6} m_{2}^{3}m_{3}^{3}

P_{510} = \frac{(s + 1)_{6}}{120} r_{1}^{5}r_{2} − \frac{(s + 1)_{5}}{24} m_{3}r_{1}^{3}(r_{1}r_{3} + 4m_{2}r_{2}) + \frac{(s + 1)_{4}}{2} m_{2}m_{3}^{2}r_{1}(r_{1}r_{3} + m_{2}r_{2})

− \frac{(s + 1)_{3}}{2} m_{2}^{2}m_{3}^{3}r_{3}

P_{420} = \frac{(s + 1)_{6}}{48} r_{1}^{4}r_{2}^{2} − \frac{(s + 1)_{5}}{24} m_{3}r_{1}^{2}(4r_{1}r_{2}r_{3} + m_{1}r_{1}^{2} + 6m_{2}r_{2}^{2})

+ \frac{(s + 1)_{4}}{4} m_{3}^{2} (r_{1}^{2}r_{3}^{2} + 4m_{2}r_{1}r_{2}r_{3} + 2m_{1}m_{2}r_{1}^{2} + m_{2}^{2}r_{2}^{2})

− \frac{(s + 1)_{3}}{2} m_{2}m_{3}^{3}(m_{1}m_{2} + r_{3}^{2})

P_{411} = \frac{(s + 2)_{4}}{24} ((s + 2)(s + 5) r_{1}^{4}r_{2}r_{3} − (s + 5) m_{1}r_{1}^{5} − 4(s + 2) r_{1}^{3}(m_{2}r_{2}^{2} + m_{3}r_{3}^{2}))

− \frac{(s + 2)_{3}}{6} (3(s + 2)_{2} m_{2}m_{3}r_{1}^{2}r_{2}r_{3} − (7s + 29)m_{1}m_{2}m_{3}r_{1}^{3}

− 6(s + 2)m_{2}m_{3}r_{1}(m_{2}r_{2}^{2} + m_{3}r_{3}^{2}))

+ \frac{(s + 1)_{3}(s + 2)}{2}m_{2}^{2}m_{3}^{2}r_{2}r_{3} − \frac{3(s + 2)_{2}(3s + 11)}{2}m_{1}m_{2}^{2}m_{3}^{2}r_{1}

P_{330} = \frac{(s + 1)_{6}}{36} r_{1}^{3}r_{2}^{3} − \frac{(s + 1)_{5}}{12} m_{3}r_{1}r_{2}(3r_{1}r_{2}r_{3} + 2m_{1}r_{1}^{2} + 2m_{2}r_{2}^{2})

+ \frac{(s + 1)_{4}}{2} m_{3}^{2}(r_{1}r_{2}r_{3}^{2} + m_{1}r_{1}^{2}r_{3} + m_{2}r_{2}^{2}r_{3} + 2m_{1}m_{2}r_{1}r_{2})

− \frac{(s + 1)_{3}}{6} m_{3}^{3}r_{3}(6m_{1}m_{2} + r_{3}^{2})

P_{321} = \frac{(s + 2)_{4}(s + 3)_{2}}{12} r_{1}^{3}r_{2}^{2}r_{3}

− \frac{(s + 2)_{3}}{12} r_{1}^{2}r_{2}(6(s + 3)_{2}m_{3}r_{3}^{2} + 3(s + 3)(s + 5)m_{2}r_{2}^{2} + 2(s + 4)_{2}m_{1}r_{1}^{2})

− \frac{(s + 2)_{3}}{6} m_{3}r_{1}r_{3}((s − 1)(s + 4)m_{1}r_{1}^{2} + 3(s + 1)(s + 3)m_{2}r_{2}^{2} − 3(s + 3)m_{3}r_{3}^{2})

+ \frac{(s + 2)_{2}}{2} m_{2}m_{3}r_{2}((s + 4)(7s + 25)m_{1}r_{1}^{2} + (s + 3)_{2}m_{2}r_{2}^{2} + (s + 3)(2s + 5)m_{3}r_{3}^{2})

+ (s + 2)_{2}(s^{2} − 13)m_{1}m_{2}m_{3}^{2}r_{1}r_{3} − (s + 2)_{2}(5s + 17)m_{1}m_{2}^{2}m_{3}^{2}r_{2}

P_{222} = \frac{(s + 3)_{2}^{3}}{8} r_{1}^{2}r_{2}^{2}r_{3}^{2} − \frac{(s + 3)^{3}(s + 4)^{2}}{2} r_{1}r_{2}r_{3}(m_{1}r_{1}^{2} + m_{2}r_{2}^{2} + m_{3}r_{3}^{2})

− \frac{(s − 2)(s + 3)^{3}(s + 4)}{4}(m_{1}m_{2}r_{1}^{2}r_{2}^{2} + m_{2}m_{3}r_{2}^{2}r_{3}^{2} + m_{1}m_{3}r_{1}^{2}r_{3}^{2})

+ \frac{(s + 3)^{2}(s + 4)^{2}}{4} (m_{1}^{2}r_{1}^{4} + m_{2}^{2}r_{2}^{4} + m_{3}^{2}r_{3}^{4}) + (s + 3)^{3}(7s + 22)m_{1}m_{2}m_{3}r_{1}r_{2}r_{3}

+ \frac{(s + 3)(s^{2} − 5s − 30)}{2} m_{1}m_{2}m_{3}((s + 3)(m_{1}r_{1}^{2} + m_{2}r_{2}^{2} + m_{3}r_{3}^{2}) − 2m_{1}m_{2}m_{3})

Table 2. Spherical polynomials and Gram matrices for n = 4 and n = 5 n = 4 .

First we repeat the example at the end of section 5. For multidegree a = (1 1 1 1), we have k ( = \frac{1}{2} ∑_{i=1}^{4} a_{i})) = 2 and dim P_{a}(d) = 3. For

A = ( \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix} ) , B = ( \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ) , C = ( \begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ) ∈ N_{0}(a),

the monomial basis is given by P_{A}^{M} = t_{12}t_{34}, P_{B}^{M} = t_{13}t_{24}, P_{C}^{M} = t_{14}t_{23}. The Gram matrix ((P_{ν ν ν}^{D}, P_{μ μ μ}^{D}))_{ν ν ν,μ μ μ∈N_{0}(a)} also tells us the coefficients of the expansion of P_{ν ν ν}^{D} in terms of the monomial basis P_{μ μ μ}^{M}, e.g. we have

d(d − 1)(d + 2) P_{A}^{D} = (d + 1)P_{A}^{M} − P_{B}^{M} − P_{C}^{M} .

If we encode the multi-indices ννν of the descending basis by the corresponding monomials x^{ν ν ν} = ∏_{i<j} x_{ij}^{ν_{ij}}, then this formula can be expressed in a visually clearer form by presenting the coefficients of the polynomial d(d − 1)(d + 2) P_{A}^{D} in the form of a table

\begin{aligned}& t_{12}t_{34} & t_{13}t_{24} & t_{14}t_{23} \\ x_{12}x_{34} & d + 1 & −1 & −1 \\ x_{13}x_{24} & −1 & d + 1 & −1 \\ x_{14}x_{23} & −1 & −1 & d + 1 \end{aligned}

and this is the form in which we will present the data for higher multi-indices.

For any polynomial P (T) = P ((t_{ij})) ∈ ℂ[S_{n}] and a permutation σ on n letters, we write P^{σ}(T) = P ((t_{σ(i),σ(j)})). If P (T) ∈ P_{a}(d), then we have P^{σ} ∈ P_{σa}(d) where σa = (a_{σ(i)}). It is obvious that (P^{σ}, Q^{σ})_{d} = (P, Q)_{d}. So if σa = a, then σ permutes the monomial basis, and also the descending basis. In the above example, we have P_{B}^{M} = P_{A}^{(23)} and P_{C} = P_{A}^{(24)}, so the symmetry of the table is quite obvious.

a = (1 1 1 3), k = 3, dim P_{a}(d) = 1 . Here, since the dimension is 1 we have to give only one monomial basis element P_{ν ν ν}^{M}. It is

t_{14}t_{24}t_{34} − \frac{t_{44}}{d + 2} (t_{12}t_{34} + t_{13}t_{24} + t_{14}t_{23}) ,

where we do not need to specify the index ννν because the leading term (the only one that survives if we set all t_{ii} equal to 0) tells us us what it is. In this case the corresponding descending basis element P_{ν ν ν}^{D} is simply 1/(d − 1)d(d + 4) times the monomial basis element, which we can express (rather unnecessarily, in this case) in our new tabular form by saying that the polynomial (d − 1)d(d + 4)P_{ν ν ν}^{D} is given by the table

\begin{aligned}& t_{14}t_{24}t_{34} \\ x_{14}x_{24}x_{34} & 1 \end{aligned}

From now on we list only the monomial basis polynomials (writing simply P_{i}, 1 ≤ i ≤ dim, rather than P_{ν ν ν}^{M}, since ννν can be read off from the leading term, and including symmetries as explained above), the common factor F by which the descending basis elements must be multiplied in order to make all coefficients in their representation with respect to the monomial basis polynomials with integral coefficients, and then a table of these coefficients.

a = (1 1 2 2), k = 3, dim P_{a}(d) = 3 .

P_{1} = t_{12}t_{34}^{2} − \frac{1}{d} t_{12}t_{33}t_{44}
P_{2} = t_{13}t_{24}t_{34} − \frac{1}{d} (t_{14}t_{24}t_{33} + t_{13}t_{23}t_{44}) + \frac{1}{d^{2}} t_{12}t_{33}t_{44}
P_{3} = (P_{2})^{(34)} = t_{14}t_{23}t_{34} − \frac{1}{d} (t_{14}t_{24}t_{33} + t_{13}t_{23}t_{44}) + \frac{1}{d^{2}} t_{12}t_{33}t_{44}
F = 2(d − 2)(d − 1)d(d + 2)(d + 4)

\begin{aligned}& t_{12}t_{34}^{2} & t_{13}t_{24}t_{34} & t_{14}t_{23}t_{34} \\ x_{12}x_{34}^{2} & d^{2} + 2d − 4 & −2d & −2d \\ x_{13}x_{24}x_{34} & −2d & 2(d^{2} + d − 4) & −2d + 8 \\ x_{14}x_{23}x_{34} & −2d & −2d + 8 & 2(d^{2} + d − 4) \end{aligned}

a = (1 1 2 4): k = 4, dim P_{a}(d) = 1.

P_{1} = t_{14}t_{24}t_{34}^{2} − \frac{t_{44}}{d + 4}(t_{14}t_{24}t_{33} + 2t_{14}t_{23}t_{34} + 2t_{13}t_{24}t_{34} + t_{12}t_{34}^{2})

+ \frac{t_{44}^{2}}{(d + 2)(d + 4)} (2t_{13}t_{23} + t_{12}t_{33})

F = 2(d − 1)d(d + 1)(d + 6) and P^{D} = F^{−1}P^{M} = F^{−1}P_{1}

a = (1 1 3 3), k = 4, dim P_{a}(d) = 3.

P_{1} = t_{12}t_{34}^{3} − \frac{3}{d + 2} t_{12}t_{34}t_{33}t_{44}
P_{2} = t_{13}t_{24}t_{34}^{2} − \frac{2}{d + 2}(t_{13}t_{23}t_{34}t_{44} + t_{14}t_{24}t_{33}t_{34})

− \frac{1}{(d + 2)^{2}} (d t_{13}t_{24}t_{33}t_{44} − 2 t_{12}t_{33}t_{34}t_{44} − 2 t_{14}t_{23}t_{33}t_{44})

P_{3} = (P_{2})^{(34)} = t_{14}t_{23}t_{34}^{2} − \frac{2}{d + 2}(t_{14}t_{24}t_{33}t_{34} + t_{13}t_{23}t_{34}t_{44})

− \frac{1}{(d + 2)^{2}} (d t_{14}t_{23}t_{33}t_{44} − 2 t_{12}t_{33}t_{34}t_{44} − 2 t_{13}t_{24}t_{33}t_{44})

F = 6(d − 2)(d − 1)d(d + 1)(d + 4)(d + 6)

\begin{aligned}& t_{12}t_{34}^{3} & t_{13}t_{24}t_{34}^{2} & t_{14}t_{23}t_{34}^{2} \\ x_{12}x_{34}^{3} & d^{2} + 5d − 2 & −3(d + 2) & −3(d + 2) \\ x_{13}x_{24}x_{34}^{2} & −3(d + 2) & 3(d^{2} + 3d − 6) & −3(d − 6) \\ x_{14}x_{23}x_{34}^{2} & −3(d + 2) & −3(d − 6) & 3(d^{2} + 3d − 6) \end{aligned}

a = (1 2 2 3), k = 4, dim P_{a}(d) = 3.

P_{1} = t_{12}t_{24}t_{34}^{2} − \frac{1}{d} t_{14}t_{22}t_{34}^{2}

+ \frac{t_{44}}{d(d + 2)}(t_{14}t_{22}t_{33} + 2 t_{13}t_{22}t_{34} − d t_{12}(t_{24}t_{33} + 2 t_{23}t_{34}))

P_{2} = P_{1}^{(23)} = t_{13}t_{24}^{2}t_{34} − \frac{1}{d} t_{14}t_{24}^{2}t_{33}

+ \frac{t_{44}}{d(d + 2)}(t_{14}t_{22}t_{33} + 2 t_{12}t_{24}t_{33} − d t_{13}(t_{22}t_{34} + 2 t_{23}t_{24}))

P_{3} = t_{14}t_{23}t_{24}t_{34} − \frac{1}{d} (t_{14}t_{24}^{2}t_{33} + t_{14}t_{22}t_{34}^{2})

− \frac{t_{44}}{d(d + 2)}(d t_{23}(t_{14}t_{23} + t_{13}t_{24} + t_{12}t_{34}) − 2 (t_{14}t_{22}t_{33} + t_{12}t_{24}t_{33} + t_{13}t_{22}t_{34}))

F = 2(d − 2)(d − 1)d(d + 1)(d + 4)(d + 6)

\begin{aligned}& t_{12}t_{24}t_{34}^{2} & t_{13}t_{24}^{2}t_{34} & t_{14}t_{23}t_{24}t_{34} \\ x_{12}x_{24}x_{34}^{2} & d^{2} + 4d − 4 & −2(d + 2) & −2(d − 2) \\ x_{13}x_{24}^{2}x_{34} & −2(d + 2) & d^{2} + 4d − 4 & −2(d − 2) \\ x_{14}x_{23}x_{24}x_{34} & −2(d − 2) & −2(d − 2) & 2(d^{2} + 3d − 2) \end{aligned}

a = (2 2 2 2), k = 4, dim P_{a}(d) = 6.

P_{1} = t_{12}^{2}t_{34}^{2} − \frac{1}{d}(t_{12}^{2}t_{33}t_{44} + t_{11}t_{22}t_{34}^{2}) + \frac{1}{d^{2}} t_{11}t_{22}t_{33}t_{44} = \frac{1}{d^{2}} (dt_{12}^{2} − t_{11}t_{22})(dt_{34}^{2} − t_{33}t_{44})
P_{2} = (P_{1})^{(23)} = t_{13}^{2}t_{24}^{2} − \frac{1}{d}(t_{13}^{2}t_{22}t_{44} + t_{24}^{2}t_{11}t_{33}) + \frac{1}{d^{2}} t_{11}t_{22}t_{33}t_{44}
P_{3} = (P_{1})^{(24)} = t_{14}^{2}t_{23}^{2} − \frac{1}{d}(t_{14}^{2}t_{22}t_{33} + t_{23}^{2}t_{11}t_{44}) + \frac{1}{d^{2}} t_{11}t_{22}t_{33}t_{44}
P_{4} = t_{13}t_{14}t_{23}t_{24} − \frac{1}{d}(t_{13}t_{14}t_{22}t_{34} + t_{12}t_{14}t_{24}t_{33} + t_{11}t_{23}t_{24}t_{34} + t_{12}t_{13}t_{23}t_{44})

+ \frac{1}{d^{2}} (t_{14}^{2}t_{22}t_{33} + t_{11}t_{24}^{2}t_{33} + t_{11}t_{22}t_{34}^{2} + t_{13}^{2}t_{22}t_{44} + t_{11}t_{23}^{2}t_{44} + t_{12}^{2}t_{33}t_{44}) − \frac{3}{d^{3}} t_{11}t_{22}t_{33}t_{44}

P_{5} = (P_{4})^{(23)} = t_{12}t_{14}t_{23}t_{34} − \frac{1}{d}(t_{13}t_{14}t_{22}t_{34} + t_{12}t_{14}t_{24}t_{33} + t_{11}t_{23}t_{24}t_{34} + t_{12}t_{13}t_{23}t_{44})

+ \frac{1}{d^{2}} (t_{14}^{2}t_{22}t_{33} + t_{11}t_{24}^{2}t_{33} + t_{11}t_{22}t_{34}^{2} + t_{13}^{2}t_{22}t_{44} + t_{11}t_{23}^{2}t_{44} + t_{12}^{2}t_{33}t_{44}) − \frac{3}{d^{3}} t_{11}t_{22}t_{33}t_{44}

P_{6} = (P_{4})^{(24)} = t_{12}t_{13}t_{24}t_{34} − \frac{1}{d}(t_{13}t_{14}t_{22}t_{34} + t_{12}t_{14}t_{24}t_{33} + t_{11}t_{23}t_{24}t_{34} + t_{12}t_{13}t_{23}t_{44})

+ \frac{1}{d^{2}} (t_{14}^{2}t_{22}t_{33} + t_{11}t_{24}^{2}t_{33} + t_{11}t_{22}t_{34}^{2} + t_{13}^{2}t_{22}t_{44} + t_{11}t_{23}^{2}t_{44} + t_{12}^{2}t_{33}t_{44}) − \frac{3}{d^{3}} t_{11}t_{22}t_{33}t_{44}

F = 4(d − 3)(d − 2)(d − 1)d(d + 1)(d + 2)(d + 4)(d + 6)

\begin{aligned}& t_{12}^{2}t_{34}^{2} & t_{13}^{2}t_{24}^{2} & t_{14}^{2}t_{23}^{2} \\ x_{12}^{2}x_{34}^{2} & d^{4} + 5d^{3} − 10d^{2} − 36d + 24 & 2(d^{2} + 12) & 2(d^{2} + 12) \\ x_{13}^{2}x_{24}^{2} & 2(d^{2} + 12) & d^{4} + 5d^{3} − 10d^{2} − 36d + 24 & 2(d^{2} + 12) \\ x_{14}^{2}x_{23}^{2} & 2(d^{2} + 12) & 2(d^{2} + 12) & d^{4} + 5d^{3} − 10d^{2} − 36d + 24 \\ x_{13}x_{14}x_{23}x_{24} & 4(d − 10)d & −4(d − 2)d(d + 4) & −4(d − 2)d(d + 4) \\ x_{12}x_{14}x_{23}x_{34} & −4(d − 2)d(d + 4) & 4(d − 10)d & −4(d − 2)d(d + 4) \\ x_{12}x_{13}x_{24}x_{34} & −4(d − 2)d(d + 4) & −4(d − 2)d(d + 4) & 4(d − 10)d \end{aligned}

\begin{aligned}& t_{13}t_{14}t_{23}t_{24} & t_{12}t_{14}t_{23}t_{34} & t_{12}t_{13}t_{24}t_{34} \\ x_{12}^{2}x_{34}^{2} & 4(d − 10)d & −4(d − 2)d(d + 4) & −4(d − 2)d(d + 4) \\ x_{13}^{2}x_{24}^{2} & −4(d − 2)d(d + 4) & 4(d − 10)d & −4(d − 2)d(d + 4) \\ x_{14}^{2}x_{23}^{2} & −4(d − 2)d(d + 4) & −4(d − 2)d(d + 4) & 4(d − 10)d \\ x_{13}x_{14}x_{23}x_{24} & 4(d − 2)^{2}(d + 3)(d + 4) & −4(d^{3} − 7d^{2} − 10d + 24) & −4(d^{3} − 7d^{2} − 10d + 24) \\ x_{12}x_{14}x_{23}x_{34} & −4(d^{3} − 7d^{2} − 10d + 24) & 4(d − 2)^{2}(d + 3)(d + 4) & −4(d^{3} − 7d^{2} − 10d + 24) \\ x_{12}x_{13}x_{24}x_{34} & −4(d^{3} − 7d^{2} − 10d + 24) & −4(d^{3} − 7d^{2} − 10d + 24) & 4(d − 2)^{2}(d + 3)(d + 4) \end{aligned}

n = 5 . a = (1 1 1 1 2). dim P_{a}(d) = 6.

In this case the symmetry group acts transitively on the monomial basis.

P_{1} = t_{12}t_{35}t_{45} − \frac{1}{d} t_{12}t_{34}t_{55}
P_{2} = (P_{1})^{(23)} = t_{13}t_{25}t_{45} − \frac{1}{d} t_{13}t_{24}t_{55}
P_{3} = (P_{1})^{(24)} = t_{14}t_{25}t_{35} − \frac{1}{d} t_{14}t_{23}t_{55}
P_{4} = (P_{1})^{(13)} = t_{15}t_{23}t_{45} − \frac{1}{d} t_{14}t_{23}t_{55}
P_{5} = (P_{1})^{(14)} = t_{15}t_{24}t_{35} − \frac{1}{d} t_{13}t_{24}t_{55}
P_{6} = (P_{1})^{(13)(24)} = t_{15}t_{25}t_{34} − \frac{1}{d} t_{12}t_{34}t_{55}
F = (d − 2)(d − 1)d(d + 2)(d + 4)

\begin{aligned}& t_{12}t_{35}t_{45} & t_{13}t_{25}t_{45} & t_{14}t_{25}t_{35} & t_{15}t_{23}t_{45} & t_{15}t_{24}t_{35} & t_{15}t_{25}t_{34} \\ x_{12}x_{35}x_{45} & d^{2} + 2d − 4 & −d & −d & −d & −d & 4 \\ x_{13}x_{25}x_{45} & −d & d^{2} + 2d − 4 & −d & −d & 4 & −d \\ x_{14}x_{25}x_{35} & −d & −d & d^{2} + 2d − 4 & 4 & −d & −d \\ x_{15}x_{23}x_{45} & −d & −d & 4 & d^{2} + 2d − 4 & −d & −d \\ x_{15}x_{24}x_{35} & −d & 4 & −d & −d & d^{2} + 2d − 4 & −d \\ x_{15}x_{25}x_{34} & 4 & −d & −d & −d & −d & d^{2} + 2d − 4 \end{aligned}

a = (1 1 1 1 4), dim P_{a}(d) = 1.

P_{1} = t_{15}t_{25}t_{35}t_{45} − \frac{t_{55}}{d + 4}(t_{12}t_{35}t_{45} + t_{13}t_{25}t_{45} + t_{14}t_{25}t_{35} + t_{15}(t_{25}t_{34} + t_{24}t_{35} + t_{23}t_{45}))

+ \frac{t_{55}^{2}}{(d + 2)(d + 4)}(t_{12}t_{34} + t_{13}t_{24} + t_{14}t_{23})

F = (d − 1)d(d + 1)(d + 6) and P^{D} = F^{−1}P^{M} = F^{−1}P_{1}

a = (1 1 1 2 3), dim P_{a}(d) = 6.

Here the monomial basis is given by P_{1}, P_{1}^{(23)}, P_{1}^{(123)}, P_{2}, P_{2}^{(12)} and P_{2}^{(13)}, where

P_{1} = t_{12}t_{35}t_{45}^{2} − \frac{1}{d + 2} t_{12}t_{55}(t_{35}t_{44} + 2 t_{34}t_{45})
P_{2} = t_{14}t_{25}t_{35}t_{45} − \frac{1}{d}t_{15}t_{25}t_{35}t_{44}

+ \frac{1}{d(d + 2)} t_{55} (t_{44}(t_{15}t_{23} + t_{13}t_{25} + t_{12}t_{35}) − d t_{14}(t_{23}t_{45} + t_{24}t_{35} + t_{25}t_{34}))

In this case we have

F = 2(d − 2)(d − 1)d(d + 1)(d + 4)(d + 6)

and the product of F with the Gram matrix of the descending basis (ordered as above) is

\begin{aligned}& t_{12}t_{35}t_{45}^{2} & t_{13}t_{25}t_{45}^{2} & t_{15}t_{23}t_{45}^{2} \\ x_{12}x_{35}x_{45}^{2} & d^{2} + 5d − 2 & −(d + 2) & −(d + 2) \\ x_{13}x_{25}x_{45}^{2} & −(d + 2) & d^{2} + 5d − 2 & −(d + 2) \\ x_{15}x_{23}x_{45}^{2} & −(d + 2) & −(d + 2) & d^{2} + 5d − 2 \\ x_{14}x_{25}x_{35}x_{45} & −2(d + 2) & −2(d + 2) & 8 \\ x_{15}x_{24}x_{35}x_{45} & −2(d + 2) & 8 & −2(d + 2) \\ x_{15}x_{25}x_{34}x_{45} & 8 & −2(d + 2) & −2(d + 2) \end{aligned}

\begin{aligned}& t_{14}t_{25}t_{35}t_{45} & t_{15}t_{24}t_{35}t_{45} & t_{15}t_{25}t_{34}t_{45} \\ x_{12}x_{35}x_{45}^{2} & −2(d + 2) & −2(d + 2) & 8 \\ x_{13}x_{25}x_{45}^{2} & −2(d + 2) & 8 & −2(d + 2) \\ x_{15}x_{23}x_{45}^{2} & 8 & −2(d + 2) & −2(d + 2) \\ x_{14}x_{25}x_{35}x_{45} & 2(d^{2} + 4d − 4) & −2(d − 2) & −2(d − 2) \\ x_{15}x_{24}x_{35}x_{45} & −2(d − 2) & 2(d^{2} + 4d − 4) & −2(d − 2) \\ x_{15}x_{25}x_{34}x_{45} & −2(d − 2) & −2(d − 2) & 2(d^{2} + 4d − 4) \end{aligned}

a = (1 1 2 2 2), dim P_{a}(d) = 10.

The monomial basis is P_{1}, P_{2}, P_{2}^{(34)}, P_{2}^{(35)}, P_{3}, P_{3}^{(45)}, P_{3}^{(34)}, P_{3}^{(345)}, P_{3}^{(354)}, P_{3}^{(35)}, with

P_{1} = t_{12}t_{34}t_{35}t_{45} − \frac{t_{12}}{d} (t_{35}^{2}t_{44} + t_{33}t_{45}^{2} + t_{34}^{2}t_{55}) + \frac{2}{d^{2}} t_{12}t_{33}t_{44}t_{55},
P_{2} = t_{13}t_{23}t_{45}^{2} − \frac{1}{d}(t_{13}t_{23}t_{44}t_{55} + t_{12}t_{33}t_{45}^{2}) + \frac{1}{d^{2}} t_{12}t_{33}t_{44}t_{55}

= \frac{1}{d^{2}} (d t_{13}t_{23} − t_{12}t_{33})(d t_{45}^{2} − t_{44}t_{55}),

P_{3} = t_{13}t_{24}t_{35}t_{45} − \frac{1}{d} (t_{13}t_{25}t_{35}t_{44} + t_{13}t_{24}t_{34}t_{55} + t_{15}t_{24}t_{33}t_{45})

+ \frac{1}{d^{2}} (t_{13}t_{23}t_{44}t_{55} + t_{14}t_{24}t_{33}t_{55} + t_{15}t_{25}t_{33}t_{44}) − \frac{1}{d^{3}} t_{12}t_{33}t_{44}t_{55}

F = 2(d − 3)(d − 2)(d − 1)d(d + 1)(d + 2)(d + 4)(d + 6)

In this case, the Gram matrix of the descending basis, multiplied by F, is a 10 × 10 matrix with rather complicated entries (for instance, its first two diagonal entries are 2(d−2)(d^{3}+6d^{2}+3d−6) and d^{4} + 5d^{3} − 10d^{2} − 36d + 24), so we do not write it out. However, the Gram matrix of the monomial basis in this case is somewhat simpler, and has polynomial coefficients after being multiplied by F^{∗} = d^{2}/(d − 1)(d + 2). We give a part of this matrix, namely, the set of values F^{∗}(P_{i}, P_{μ μ μ}^{M}) for 1 ≤ i ≤ 3 and all ten μ μ μ (ordered as above), in the following table. The remaining 70 values, of course, follow from the symmetry.

\begin{aligned}& t_{12}t_{34}t_{35}t_{45} & t_{13}t_{23}t_{45}^{2} & t_{14}t_{24}t_{35}^{2} & t_{15}t_{25}t_{34}^{2} \\ P_{1} & (d − 2)d^{2}(d + 4) & 0 & 0 & 0 \\ P_{2} & 0 & 2(d − 1)d^{2}(d + 2) & 4d^{2} & 4d^{2} \\ P_{3} & (d − 2)d(d + 4) & 2(d − 1)d(d + 2) & 2(d − 1)d(d + 2) & 2(d − 2)d \end{aligned}

\begin{aligned}& t_{13}t_{24}t_{35}t_{45} & t_{13}t_{25}t_{34}t_{45} & t_{14}t_{23}t_{35}t_{45} \\ P_{1} & (d − 2)d(d + 4) & (d − 2)d(d + 4) & (d − 2)d(d + 4) \\ P_{2} & 2(d − 1)d(d + 2) & 2(d − 1)d(d + 2) & 2(d − 1)d(d + 2) \\ P_{3} & (d − 1)^{2}(d + 2)^{2} & (d − 2)(d − 1)(d + 2) & d^{3} + 3d^{2} − 4d + 4 \end{aligned}

\begin{aligned}& t_{14}t_{25}t_{34}t_{35} & t_{15}t_{23}t_{34}t_{45} & t_{15}t_{24}t_{34}t_{35} \\ P_{1} & (d − 2)d(d + 4) & (d − 2)d(d + 4) & (d − 2)d(d + 4) \\ P_{2} & 2(d − 2)d & 2(d − 1)d(d + 2) & 2(d − 2)d \\ P_{3} & (d − 2)^{2} & (d − 2)^{2} & (d − 2)(d − 1)(d + 2) \end{aligned}

Table 3. Determinants of Gram matrices

In §9.C.8. we explained that the determinant GD_{a}^{M}(d) of the Gram matrix for the monomial basis has the form C ∏_{m≥1−n}(d + m)^{e(m)} for some C ∈ ℚ^{×} and e(m) ∈ ℤ. We give a table of the exponents e(m) and of the constant C for n = 4 and all tuples a with k ≤ 8 (written as (a_{1} a_{2} a_{3} a_{4}) with 1 ≤ a_{1} ≤ a_{2} ≤ a_{3} ≤ a_{4}, and also a_{4} ≤ a_{1} + a_{2} + a_{3} since otherwise the dimension N_{0}(a) is 0), and then a shorter table for n = 5 and all tuples a with k ≤ 6. The tabulated values of m are consecutive to the left of the vertical line and even to its right.

n = 4 . k = 2, 3, 4, 5 .

\begin{aligned}k & dim P_{a}(d) & a & C & −3 & −2 & −1 & 0 & 1 & 2 & 4 & 6 & 8 \\ 2 & 3 & (1111) & 1 & & & 2 & 3 & 0 & 1 \\ 3 & 1 & (1113) & 1 & & & 1 & 1 & 0 & 0 & 1 \\ & 3 & (1122) & 2 & & 1 & 3 & 1 & 0 & 3 & 1 \\ 4 & 1 & (1124) & 2 & & & 1 & 1 & 1 & 0 & 0 & 1 \\ & 3 & (1133) & 2^{3} · 3 & & 1 & 3 & 3 & 2 & −1 & 3 & 1 \\ & & (1223) & 2^{2} & & 2 & 3 & 1 & 2 & 0 & 3 & 1 \\ & 6 & (2222) & 2^{6} & 1 & 3 & 6 & 1 & 3 & 6 & 3 & 1 \\ 5 & 1 & (1135) & 2 · 3 & & & 1 & 1 & 1 & 1 & 0 & 0 & 1 \\ & & (1225) & 2^{2} & & & 1 & 1 & 1 & 1 & 0 & 0 & 1 \\ & 3 & (1144) & 2^{5} · 3^{3} & & 1 & 3 & 3 & 3 & 2 & −1 & 3 & 1 \\ & & (1234) & 2^{4} · 3 & & 2 & 3 & 2 & 3 & 1 & 0 & 3 & 1 \\ & & (1333) & 2^{3} & & 3 & 3 & 3 & 3 & −4 & 3 & 3 & 1 \\ & & (2224) & 2^{3} & & 3 & 3 & 0 & 3 & 2 & 0 & 3 & 1 \\ & 6 & (2233) & 2^{8} · 3 & 1 & 4 & 6 & 4 & 5 & 0 & 6 & 3 & 1 \end{aligned}

k = 6 .

\begin{aligned}dim & a & C & −3 & −2 & −1 & 0 & 1 & 2 & 3 & 4 & 6 & 8 & 10 \\ 1 & (1146) & 2^{3} · 3 & & & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 \\ & (1236) & 2^{2} · 3 & & & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 \\ & (2226) & 2^{3} & & & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 \\ 3 & (1155) & 2^{9} · 3^{3} · 5 & & 1 & 3 & 3 & 3 & 3 & 2 & 0 & −1 & 3 & 1 \\ & (1245) & 2^{6} · 3^{3} & & 2 & 3 & 2 & 3 & 3 & 2 & −1 & 0 & 3 & 1 \\ & (1335) & 2^{6} · 3^{2} & & 2 & 3 & 3 & 3 & 1 & 2 & 0 & 0 & 3 & 1 \\ & (1344) & 2^{5} · 3 & & 3 & 3 & 4 & 3 & −1 & 2 & −3 & 3 & 3 & 1 \\ & (2235) & 2^{5} · 3 & & 3 & 3 & 1 & 3 & 2 & 2 & 0 & 0 & 3 & 1 \\ 6 & (2244) & 2^{13} · 3^{3} & 1 & 4 & 6 & 5 & 6 & 4 & 3 & −3 & 6 & 3 & 1 \\ & (2334) & 2^{10} · 3^{2} & 1 & 5 & 6 & 6 & 6 & −1 & 3 & 0 & 6 & 3 & 1 \\ 10 & (3333) & 2^{18} · 3^{6} & 3 & 7 & 10 & 11 & 9 & −4 & 4 & 10 & 6 & 3 & 1 \end{aligned}

k = 7 .

dim  a  C  −3 −2 −1 0 1 2 3 4 6 8 10 12
1  ⋆  (see below)  1 1 1 1 1 1 0 0 0 1
3 (1166) 2^{10} · 3^{4} · 5^{3}  1 3 3 3 3 3 2 0 −1 3 1
(1256) 2^{10} · 3^{3} · 5  2 3 2 3 3 3 2 −1 0 3 1
(1346)  2^{8} · 3^{4}  2 3 3 3 2 3 1 0 0 3 1
(1355)  2^{7} · 3^{3}  3 3 4 3 0 3 1 −3 3 3 1
(1445)  2^{7} · 3^{2}  3 3 5 3 1 3 −4 0 3 3 1
(2246)  2^{7} · 3^{3}  3 3 1 3 3 3 1 0 0 3 1
(2336)  2^{7} · 3^{2}  3 3 2 3 1 3 2 0 0 3 1
6 (2255) 2^{17} · 3^{6} · 5 1 4 6 5 6 6 5 2 −3 6 3 1
(2345) 2^{15} · 3^{4}  1 5 6 7 6 2 5 0 0 6 3 1
(2444) 2^{12} · 3^{3}  1 6 6 8 6 3 5 −9 6 6 3 1
(3335) 2^{12} · 3^{3}  1 6 6 8 6 −3 5 3 0 6 3 1
10 (3344) 2^{23} · 3^{8}  3 8 10 13 10 1 7 −2 10 6 3 1

Here ⋆ denotes the list {(1157), (1247), (1337), (2237)} of a with dim P_{a}(d) = 1, the corresponding C-values being 120, 48, 36 and 24.

k = 8 .

dim a  C  −3 −2 −1 0 1 2 3 4 5 6 8 10 12 14
1  ⋆  (see below)  1 1 1 1 1 1 1 0 0 0 0 1
3 (1177) 2^{12} · 3^{6} · 5^{3} · 7  1 3 3 3 3 3 3 2 0 0 −1 3 1
(1267) 2^{11} · 3^{4} · 5^{3}  2 3 2 3 3 3 3 2 0 −1 0 3 1
(1357) 2^{12} · 3^{4} · 5  2 3 3 3 2 3 3 2 −1 0 0 3 1
(1366) 2^{11} · 3^{3} · 5  3 3 4 3 0 3 3 2 −1 −3 3 3 1
(1447)  2^{10} · 3^{6}  2 3 3 3 3 3 1 2 0 0 0 3 1
(1456)  2^{9} · 3^{4}  3 3 5 3 2 3 −1 2 −3 0 3 3 1
(1555)  2^{9} · 3^{3}  3 3 6 3 3 3 −3 2 −6 3 3 3 1
(2257) 2^{11} · 3^{3} · 5  3 3 1 3 3 3 3 2 −1 0 0 3 1
(2347)  2^{9} · 3^{4}  3 3 2 3 2 3 2 2 0 0 0 3 1
(3337)  2^{9} · 3^{3}  3 3 3 3 0 3 3 2 0 0 0 3 1
6 (2266) 2^{24} · 3^{7} · 5^{3} 1 4 6 5 6 6 6 5 3 −1 −3 6 3 1
(2356) 2^{19} · 3^{7} · 5 1 5 6 7 6 3 6 4 3 −3 0 6 3 1
(2446)  2^{20} · 3^{6}  1 5 6 8 6 4 6 −1 3 0 0 6 3 1
(2455)  2^{17} · 3^{5}  1 6 6 9 6 5 6 −4 3 −6 6 6 3 1
(3346)  2^{17} · 3^{5}  1 6 6 9 6 −1 6 2 3 0 0 6 3 1
10 (3445) 2^{28} · 3^{10}  3 9 10 15 10 5 9 −5 4 0 10 6 3 1
(3355) 2^{31} · 3^{11} · 5 3 8 10 14 10 4 9 4 4 −6 10 6 3 1
15 (4444) 2^{48} · 3^{18}  6 12 16 21 15 10 12 −12 5 15 10 6 3 1

Here ⋆ denotes the list {(1168), (1258), (1348), (2248), (2338)} of a with dim P_{a}(d) = 1, the corresponding C-values being 720, 240, 144, 96 and 72. n = 5 . k = 3, 4, 5, 6 .

\begin{aligned}k & dim P_{a}(d) & a & C & −4 & −3 & −2 & −1 & 0 & 1 & 2 & 3 & 4 & 6 & 8 & 10 \\ 3 & 6 & (11112) & 1 & & & 2 & 6 & 3 & 0 & 6 & 0 & 1 \\ 4 & 1 & (11114) & 1 & & & & 1 & 1 & 1 & 0 & 0 & 0 & 1 \\ & 6 & (11123) & 2^{3} & & & 3 & 6 & 5 & 3 & 0 & 0 & 6 & 1 \\ & 10 & (11222) & 2^{3} & & 1 & 7 & 10 & 1 & 4 & 10 & 0 & 6 & 1 \\ 5 & 1 & (11125) & 2 & & & & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 \\ & 6 & (11134) & 2^{6} · 3^{3} & & & 3 & 6 & 6 & 6 & 2 & 0 & 0 & 6 & 1 \\ & & (11224) & 2^{6} & & & 4 & 6 & 4 & 6 & 3 & 0 & 0 & 6 & 1 \\ & 10 & (11233) & 2^{8} · 3 & & 1 & 8 & 10 & 7 & 9 & −2 & 0 & 10 & 6 & 1 \\ & 15 & (12223) & 2^{12} & & 3 & 12 & 15 & 6 & 12 & 5 & 0 & 15 & 6 & 1 \\ & 22 & (22222) & 2^{10} & 1 & 6 & 22 & 22 & −7 & 16 & 22 & 0 & 21 & 6 & 1 \\ 6 & 1 & (11136) & 2 · 3 & & & & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 \\ & 1 & (11226) & 2^{2} & & & & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 \\ & 6 & (11145) & 2^{12} · 3^{6} & & & 3 & 6 & 6 & 6 & 6 & 3 & −1 & 0 & 6 & 1 \\ & & (11235) & 2^{9} · 3^{3} & & & 4 & 6 & 5 & 6 & 5 & 3 & 0 & 0 & 6 & 1 \\ & & (12225) & 2^{9} & & & 5 & 6 & 3 & 6 & 6 & 3 & 0 & 0 & 6 & 1 \\ & 10 & (11244) & 2^{14} · 3^{6} & & 1 & 8 & 10 & 8 & 10 & 8 & 4 & −6 & 10 & 6 & 1 \\ & & (11334) & 2^{13} · 3^{2} & & 1 & 9 & 10 & 12 & 10 & −3 & 4 & 0 & 10 & 6 & 1 \\ & 15 & (12234) & 2^{19} · 3^{3} & & 3 & 13 & 15 & 11 & 15 & 6 & 5 & 0 & 15 & 6 & 1 \\ & 21 & (22224) & 2^{30} & & 6 & 17 & 21 & 12 & 21 & 15 & 6 & 0 & 21 & 6 & 1 \\ & & (12333) & 2^{27} · 3^{3} & & 6 & 18 & 21 & 21 & 20 & −9 & 6 & 21 & 15 & 6 & 1 \\ & 29 & (22233) & 2^{31} · 3 & 1 & 10 & 28 & 29 & 15 & 28 & −1 & 7 & 29 & 21 & 6 & 1 \end{aligned}

Final remark. In these tables, non-zero exponents e(m) occur only if 1 − n ≤ m ≤ k − 3 or if k − 2 ≤ m ≤ 2k − 2 and m is even, the latter statement being highlighted by the vertical lines that we have inserted. This suggests that the dimension of the space P_{a}(d) has its generic value N_{0}(a), and that both the generic monomial and descending basis elements give bases of this space, in any of the following four cases:

• d ∉ ℤ ,

• d ≥ n,
• d ≤ −2k,
• d ≤ 2 − k and d odd.

This is in fact true, and follows by combining results given at various points of our paper, namely Theorem 3 of §3 (which tells us that the inner product of monomials is polynomial in d), Theorem 5 of §5 (which tells us how to get the monomial basis by iterating the “raising operator”), Theorem 6 of §5 (giving the duality between the monomial and descending bases), equation (117) in §9.A (which tells us how to get the generating function by repeated applications of appropriate differential operators), and equation (134) of §9.C (which relates the generating function basis to the descending basis). But there are still many mysteries, e.g., why the value of e(2k − 2) should always be 1, as the tables suggest.

List of Notations

We list the principal notations, omitting those that are used only locally, and including section or equation numbers when this seems helpful.

S_{n}  space of real symmetric n × n matrices, with coordinate T = (t_{ij})
ℂ[S_{n}], ℂ[V]  space of polynomials on S_{n} or on any vector space V
β_{n}  map (ℝ^{d})^{n} → S_{n} defined by (x_{1}, …, x_{n}) ↦ ( (x_{i}, x_{j}) ) or X ↦ XX^{t}
β_{n}^{∗}  induced map P ↦ P̃ = P ∘ β_{n} from ℂ[S_{n}] to C[(ℝ^{d})^{n}]
N  set of symmetric n × n matrices ννν = (ν_{ij}) with ν_{ij} ∈ ℤ_{≥0}, ν_{ii} even
T^{ν ν ν} (ννν ∈ N)  monomial ∏ t_{ij}^{ν_{ij}/2} in ℂ[S_{n}]
∂_{ij} (1 ≤ i, j ≤ n)  partial derivative w.r.t. t_{ij}, multiplied by 2 if i = j
0, 1, 2  vectors consisting of n 0’s, resp. 1’s, resp. 2’s
N_{0}  set of ννν ∈ N with ν_{ii} = 0 for all i
N (a), N_{0}(a) (a ∈ ℤ_{≥0}^{n})  set of ννν in N, resp. N_{0}, with ν · 1 = a
N(a), N_{0}(a) (a ∈ ℤ_{≥0}^{n})  cardinality of N (a), resp. N_{0}(a)
ℂ[S_{n}]_{a} (a ∈ ℤ_{≥0}^{n})  subspace of ℂ[S_{n}] spanned by monomials T^{ν ν ν} with ννν ∈ N (a)
D_{i}, E_{i} (1 ≤ i ≤ n)  ith Laplace, resp. Euler differential operators on ℂ[S_{n}] [ §1 ]
P^{(n)}(d), P_{a}(d) (a ∈ ℤ_{≥0}^{n})  subspace of ℂ[S_{n}], resp. ℂ[S_{n}]_{a}, annihilated by all D_{i}
S_{n}^{0}  space of matrices T ∈ S_{n} with all t_{ii} equal to 0
Φ  the generically bijective map P^{(n)}(d) ⊂ ℂ[S_{n}] → ℂ[S_{n}^{0}] ≅ ℂ^{N_{0}}
Ξ(a) (a ∈ ℤ_{≥0}^{n})  exceptional set 2ℤ ∩ ∪_{i=1}^{n}[4 − 2a_{i}, 2 − a_{i}] in Theorem 1 [ §2 ]
δ(T)^{m} (m ∈ ℤ_{≥0}^{n})  diagonal monomial ∏_{i=1}^{n} t_{ii}^{n} ∈ ℂ[S_{n}]_{2m}
P_{ν ν ν}^{M} = P_{ν ν ν,d}^{M}  monomial basis of P^{(n)}(d) [ §2 ]
H(ℝ^{d})  space Ker(Δ) of harmonic polynomials in ℂ[ℝ^{d}]
(x)_{n} (x ∈ ℂ, n ∈ ℤ_{≥0})  increasing Pochhammer symbol x(x + 1) ⋯ (x + n − 1)
π^{(d)}, Π_{a}^{(d)}  projection map ℂ[ℝ^{d}] → H(ℝ^{d}), resp. ℂ[S_{n}]_{a} → P_{a}(d) [ §2 ]
S_{n}^{1}  space of matrices T ∈ S_{n} with all t_{ii} equal to 1
S_{n}^{+}, S_{n}^{1,+}  space of positive definite matrices in S_{n}, resp. S_{n}^{1}
E_{ℝ^{d}}, E_{S^{d−1}}, E_{(ℝ^{d})^{n}}, E_{(S^{d−1})^{n}} expectation value w.r.t. probability measures on these spaces [ §3 ]
( , )_{d}, ( , )_{d}^{1}  scalar products on ℂ[S_{n}], resp. ℂ[S_{n}^{1}] [ §3 ]
E_{d}, E_{d}^{1}  the corresponding functionals ( , 1)_{d}, ( , 1)_{d}^{1}
c_{n}(d), c_{n}^{1}(d)  normalizing constants in the definitions of the scalar products
ε_{a}(d), ε_{a}(d)  normalizing factors relating the two scalar products [ eqs. (14), (18) ]
𝔏_{n}^{+}  space of lower triangular n × n matrices with positive diagonal entries
E_{d}^{𝔏}  functional on ℂ[𝔏_{n}]^{ev} defined in (25)
D_{ij}, E_{ij}, F_{ij} (1 ≤ i, j ≤ n) mixed Laplace and Euler operators and multiplication by t_{ij} [ §4 ]
𝔤  the Lie algebra spanned by all D_{ij}, E_{ij}, F_{ij} [ §4 ]
R_{ij}, C_{ij} (1 ≤ i, j ≤ n)  raising, resp. “mixed Casimir” operators on ℂ[S_{n}] [ §5 ]
P_{ν ν ν}^{D} = P_{ν ν ν,d}^{D}  descending basis of P^{(n)}(d) (§5)
K  the field ℚ(d), with d considered as a variable [ §5, §9 ]
R  the subring ℚ[d, d^{−1}, (d ± 1)^{−1}, (d ± 2)^{−1}, …] of K [ §9, eq. (112) ]
L_{p}  the differential operator w.r.t. σ_{1}, …, σ_{n} [ §9, eq. (102) ]
M_{p}  the “main” part of L_{p} [ §9, eq. (113) ]
G^{(n)}  the standard generating function [ §9, eqs. (117) and (130) ]
G_{k}  the weight k part of the generating function [ §9 ]
P_{ν ν ν}(T), P_{ν ν ν,d}(T)  the standard spherical polynomial [ §8.A, §9.C.1 ]

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Department of Mathematics, Graduate School of Science Osaka University, Toyonaka, Osaka, 560-0043, Japan E-Mail: ibukiyam@math.sci.osaka-u.ac.jp

Max-Planck Institute für Mathematik Bonn 53111, Germany and Collège de France 75005 Paris, France E-Mail: don.zagier@mpim-bonn.mpg.de