Some properties of hypergeometric series associated with mirror symmetry

Don Zagier and Aleksey Zinger

March 18, 2008

Abstract

We show that certain hypergeometric series used to formulate mirror symmetry for Calabi-Yau hypersurfaces, in string theory and algebraic geometry, satisfy a number of interesting properties. Many of these properties are used in separate papers to verify the BCOV prediction for the genus one Gromov-Witten invariants of a quintic threefold and more generally to compute the genus one Gromov-Witten invariants of any Calabi-Yau projective hypersurface.

1. Introduction

An astounding prediction for the genus zero Gromov-Witten invariants of (counts of rational curves in) a quintic threefold was made in [CaDGP]. It was formulated in terms of the function F defined in (1) below and related objects. This 1991 mirror symmetry prediction was mathematically verified about five years later. The 1993 mirror symmetry prediction of [BCOV] for the genus one Gromov-Witten invariants of a quintic threefold was recently verified in [Z1]. A generalization of this prediction for a degree n hypersurface X_{n} in ℂP^{n−1}, for an arbitrary n, is proved in [Z2]; X_{5} is a quintic threefold. The proofs in these two papers make use of the properties of F described by Theorems 1–3 below. Theorem 4 explores related properties of F; they appear to be of interest in their own right and may also be of use in computation of higher genus Gromov-Witten invariants. Some further conjectural properties are stated in Section 3.

We denote by

P ⊂ 1 + xℚ(w)[[x]]

the subgroup of power series in x with constant term 1 whose coefficients are rational functions in w which are holomorphic at w =0. Thus, the evaluation map

P → 1 + xℚ[[x]], F (w, x) ↦ F (0, x) ,

is well-defined. We define a map M : P → P by

MF (w, x) = \{1 + \frac{x}{w} \frac{∂}{∂x}\}\frac{F (w, x)}{F (0, x)} .

Our first result says that the hypergeometric functions arising in the mirror symmetry predictions are periodic fixed points of the map M.

Theorem 1. Let n be a positive integer and F ∈ P the hypergeometric series

F(w, x) = ∑_{d=0}^{∞} x^{d} \frac{∏_{r=1}^{r=nd}(nw+r)}{∏_{r=1}^{r=d}((w+r)^{n}−w^{n})} .  (1)

Then M^{n}F = F .

Note that we consider n as fixed and therefore omit it from the notations.

If we now define further power series F_{p} ∈ P and I_{p} ∈ 1 + xℚ[[x]] for all p ≥ 0 by

F_{p}(w, x) = M^{p}F(w, x) , I_{p}(x) = F_{p}(0, x) ,

so that F_{p+1} = (1 + w^{−1}x d/dx)(F_{p}/I_{p}), then Theorem 1 says that F_{n+p} = F_{p} and consequently I_{n+p} = I_{p} for all p ≥ 0. The next result gives further properties of the functions {I_{p}}_{p∈ℤ/nℤ}.

Theorem 2. The power series I_{p}(x), 0 ≤ p ≤ n − 1, satisfy

I_{0}(x) I_{1}(x) ⋯ I_{n−1}(x) = (1 − n^{n}x)^{−1},  (2)
I_{0}(x)^{n−1}I_{1}(x)^{n−2} ⋯ I_{n−1}(x)^{0} = (1 − n^{n}x)^{−(n−1)/2} ,  (3)
I_{p}(x) = I_{n−1−p}  (0 ≤ p ≤ n−1) .  (4)

We note that (2) and the symmetry property (4) imply (3). However, (3) is simpler to prove directly than (4) and will be verified together with (2) before we give the proof of (4).

The power series I_{p} describe the structure of F at w =0. We will also describe some of its structure at w =∞. We begin with the following observation, which will be proved in Subsection 2.3.

Lemma 1. If F ∈ P and M^{k}F = F for some k > 0, then every coefficient of the power series log F (w, x) ∈ ℚ(w)[[x]] is O(w) as w → ∞.

Applying this lemma to F = F, which satisfies its hypothesis by Theorem 1, we find that log F(w, x) has an asymptotic expansion ∑_{j=−1}^{∞} μ_{j}(x)w^{−j} with μ_{j}(x) ∈ xℚ[[x]] for all j ≥ −1 or equivalently, that F(w, x) itself has an asymptotic expansion

F(w, x) ∼ e^{μ(x)w} ∑_{s=0}^{∞} Φ_{s}(x) w^{−s}  (w → ∞)  (5)

for some power series μ = μ_{−1}, Φ_{0} = e^{μ_{0}}, Φ_{1} = Φ_{0}μ_{1}, … in ℚ[[x]].

Theorem 3. The first three coefficients μ(x), Φ_{0}(x), and Φ_{1}(x) in the expansion (5) are given by

μ(x) = ∫_{0}^{x} \frac{L(u) − 1}{u} du , Φ_{0}(x) = L(x) , Φ_{1}(x) = \frac{(n−2)(n+1)}{24n} (L(x) − L(x)^{n}) ,  (6)

where L(x) denotes the power series (1 − n^{n}x)^{−1/n} ∈ ℤ[[x]] .

The proof of this theorem in Subsection 2.3 can be systematized and streamlined to obtain an algorithm for computing every Φ_{s} by a differential recursion, which we now state. For integers m ≥ j ≥ 0 (and for our fixed integer n) we define H_{m,j} = H_{m,j}(X) ∈ ℚ[X] inductively by

H_{0,j} = δ_{0,j},  H_{m,j} = H_{m−1,j} + (X − 1)(X \frac{d}{dX} + \frac{m − j}{n} )H_{m−1,j−1} for m ≥ 1.  (7)

For example, for 0 ≤ j ≤ 2 we find

\begin{aligned}H_{m,0}(X) & = 1,  H_{m,1}(X) = \frac{1}{n}\binom{m}{2}(X − 1), \\ H_{m,2}(X) & = \frac{1}{n^{2}} \binom{m}{3}((n + 1)X − 1)(X − 1) + \frac{3}{n^{2}} \binom{m}{4}(X − 1)^{2} . \end{aligned}  (8)

For fixed j ≥ 1 and varying m, H_{m,j} has the form ∑_{k=1}^{j} \binom{m}{j+k}Q_{j,k}(X) with Q_{j,k} ∈ ℤ[n^{−1}, X] defined inductively by

Q_{0,k} = δ_{0,k}, Q_{j,k} = (X − 1)(XQ′_{j−1,k} + (k Q_{j−1,k} + (k + j − 1) Q_{j−1,k−1})/n for j ≥ 1.

We then define differential operators 𝔏_{k} (0 ≤ k ≤ n) on ℚ[[x]] by

𝔏_{k} = ∑_{i=0}^{k} (\binom{n}{i}H_{n−i,k−i}(L^{n}) − (L^{n} − 1) ∑_{r=1}^{k−i} \binom{n − r}{i}\frac{S_{r}(n)}{n^{r}} H_{n−i−r,k−i−r}(L^{n}))D^{i} ,  (9)

where D = x d/dx and S_{r}(n) denotes the rth elementary symmetric function of 1, 2, …, n (a Stirling number of the first kind). Using (8), we find that the first two of these operators are

𝔏_{1} = nD − (L^{n} − 1) = nLDL^{−1} ,  (10)
𝔏_{2} = \binom{n}{2}D^{2} − \frac{3(n − 1)}{2}(L^{n} − 1)D + \frac{n − 1}{n}(\frac{(n − 2)(n − 11)}{24}L^{n} − 1)(L^{n} − 1) .  (11)

Theorem 4. (i) The power series Φ_{s} ∈ ℚ[[x]], s ≥ 0, are determined by the first-order ODEs

𝔏_{1}(Φ_{s}) + \frac{1}{L} 𝔏_{2}(Φ_{s−1}) + \frac{1}{L^{2}} 𝔏_{3}(Φ_{s−2}) + ⋯ + \frac{1}{L^{n−1}} 𝔏_{n}(Φ_{s+1−n}) = 0, s ≥ 0,  (12)

(with the convention Φ_{r} = 0 for r < 0) together with the initial condition Φ_{s}(0) = δ_{0,s}. (ii) For fixed s and n, Φ_{s}(x) belongs to L ℚ[L] . (iii) For fixed s, Φ_{s}(x) belongs to ℚ(n)[L, L^{−1}, L^{n}] .

For example, from (12) for s = 0 and s = 1 together with equations (10) and (11) one finds the second and third identity in (6), and continuing the same way one obtains

Φ_{2} = \frac{(n + 1)^{2}(n − 2)^{2}}{2 (24n)^{2}} (L − 2L^{n} + L^{2n−1}) = Φ_{1}^{2}/2LΦ_{0} ,
Φ_{3} = \frac{(n + 1)(n − 2)}{30 (24n)^{3}} \{(1003n^{4} − 2366n^{3} + 3759n^{2} − 1676n − 164) L^{3n−2}
− 72 (n − 1)(3n − 1)(7n^{2} − 9n + 14) L^{2n−2}
+ 15 (n + 1)^{2}(n − 2)^{2}(L^{2n−1} − L^{n})
+ 72 (n − 1)(7n^{3} − 17n^{2} + 22n − 24) L^{n−2}
+ (5n^{4} + 134n^{3} − 447n^{2} + 308n − 556) L} .

illustrating parts (ii) and (iii) of the theorem. These expressions, and the similar formulas obtained for s ≤ 7, suggest that in fact Φ_{s} for s fixed and n varying is an element of ℚ[n, n^{−1}, L, L^{−1}, L^{n}], sharpening statement (iii), but we do not know how to prove this. Some further data and a further conjecture concerning the functions Φ_{s}(x) is given in Section 3.

2. Proofs

2.1. Preliminaries

It will be convenient to introduce notations D and D_{w} for the first order differential operators D = x \frac{d}{dx} and D_{w} = D + w on ℚ(w)[[x]]. (Here we think of w as a parameter rather than a variable and write simply \frac{d}{dx} instead of \frac{∂}{∂x}.) The effect of D_{w} on a power series ∑ c_{d}(w)x^{d} ∈ ℚ(w)[[x]] is to multiply each c_{d}(w) by w + d, so D_{w} has an inverse operator D_{w}^{−1} which replaces each c_{d}(w) by (w + d)^{−1}c_{d}(w) . The operator M defined above can be written in terms of D_{w} as F (w, x) ↦ w^{−1}D_{w}[F (w, x)/F (0, x)].

We remark that instead of working with the functions F_{p}(w, x), we could have worked with the functions R_{p}(w, t) = e^{wt}F_{p}(w, e^{t}), which are the objects that actually arise in the analysis of the mirror symmetry predictions for Gromov-Witten invariants. If we had done that, then the differential operator D_{w} = w + x d/dx would have been replaced by the simpler differential operator d/dt, explaining why this operator plays such a ubiquitous role in our analysis. But it is easier, both in the calculations and for purposes of exposition, to work with power series over ℚ(w) in a single variable x rather than with objects in the less familiar space e^{wt}ℚ(w)[[e^{t}]].

The following lemma and its corollary are the key to the proofs of the four theorems stated above.

Lemma 2. Suppose c_{0}, …, c_{m}, f, g, a are functions of t (with f not identically 0) satisfying

\begin{matrix}c_{m} f^{(m)} + c_{m−1} f^{(m−1)} + … + c_{0} f = 0 , \\ c_{m} g^{(m)} + c_{m−1} g^{(m−1)} + … + c_{0} g = a , \end{matrix}  (13)

where f^{(k)} = d^{k}f/dt^{k}. Then the function h := (g/f)′ satisfies

c̃_{m−1} h^{(m−1)} + c̃_{m−2} h^{(m−2)} + … + c̃_{0} h = a,  (14)

where c̃_{s}(t) = ∑_{r=s+1}^{m} \binom{r}{s+1} c_{r}(t) f^{(r−1−s)}(t) .

Proof: Using Leibnitz’s rule and (13), we find

a = ∑_{r=0}^{m} c_{r} (f · g/f)^{(r)} = ∑_{r=0}^{m} c_{r} (f^{(r)}g/f + ∑_{s=0}^{r−1} \binom{r}{s + 1}f^{(r−1−s)}h^{(s)}) = ∑_{s=0}^{m−1} c̃_{s} h^{(s)} .

Corollary 1. Suppose F (w, x) ∈ P satisfies

(∑_{r=0}^{m} C_{r}(x) D_{w}^{r})F (w, x) = A(w, x)  (15)

for some power series C_{0}(x), …, C_{m}(x) ∈ ℚ[[x]] and A(w, x) ∈ ℚ(w)[[x]] with A(0, x) ≡ 0. Then

(∑_{s=0}^{m−1} C̃_{s}(x) D_{w}^{s})MF (w, x) = \frac{1}{w} A(w, x) ,  (16)

where C̃_{s}(x) := ∑_{r=s+1}^{m} \binom{r}{s+1} C_{r}(x) D^{r−1−s}F (0, x) .

Proof: Apply the lemma with c_{r}(t) = C_{r}(e^{t}), f(t) = F (0, e^{t}), g(t) = e^{wt}F (w, e^{t}), a(t) = e^{wt}A(w, e^{t}), noting that then h(t) = we^{wt}MF (w, e^{t}).

2.2. Proof of Theorem 2

For the proof of (2) and (3), it is convenient to define F_{p}(w, x) also for p = −1. Set

F_{−1}(w, x) = w D_{w}^{−1}F(w, x) = ∑_{d=0}^{∞} x^{d} \frac{∏_{r=0}^{r=nd−1}(nw+r)}{∏_{r=1}^{r=d} ((w+r)^{n}−w^{n})} ∈ P .  (17)

We have F_{−1}(0, x)=1 and w^{−1}D_{w}F_{−1} =F, so F_{p} = M^{p+1}F_{−1} for all p ≥ 0, justifying the notation. It is straightforward to check that F_{−1} is a solution of the differential equation

(D_{w}^{n} − x ∏_{j=0}^{n−1}(nD_{w} + j)) F_{−1} = w^{n} F_{−1} .  (18)

This has the form of (15) with F = F_{−1}, A = w^{n}F_{−1}, m = n, and

C_{n}(x) = 1 − n^{n}x , C_{r}(x) = −n^{r} S_{n−r}(n − 1) x  (0 < r < n),  C_{0}(x) = 0 ,  (19)

where S_{n−r}(n − 1) as before denotes the (n − r)-th elementary symmetric function of 1, 2, …, n − 1. Applying Corollary 1 repeatedly, we obtain

∑_{s=0}^{n−1−p} C_{s}^{(p)}(x) D_{w}^{s}F_{p}(w, x) = w^{n−p−1} F_{−1}(w, x)  (0 ≤ p ≤ n − 1),  (20)

where C_{s}^{(0)}(x) = C_{s+1}(x) with C_{r}(x) as in (19) and C_{s}^{(p)} for p > 0 is given inductively by

C_{s}^{(p)} = ∑_{r=s+1}^{n−p} \binom{r}{s + 1} C_{r}^{(p−1)}(x) D^{r−1−s}I_{p−1}(x) .  (21)

In particular, by induction on p we find that the first two coefficients in (20) are given by

C_{n−1−p}^{(p)} = (1 − n^{n}x) ∏_{r=0}^{p−1} I_{r}(x) ,  (22)

C_{n−2−p}^{(p)} = (−\frac{n^{n}(n − 1)}{2} x + (1 − n^{n}x) ∑_{r=0}^{p−1}(n − r − 1) \frac{I′_{r}(x)}{I_{r}(x)}) ∏_{r=0}^{p−1} I_{r}(x) .  (23)

Equations (20) and (22) for p = n − 1 give

(1 − n^{n}x) ∏_{r=0}^{n−2} I_{r}(x) F_{n−1}(w, x) = F_{−1}(w, x) .  (24)

Setting w = 0 in this relation and using F_{−1}(0, x) = 1 gives equation (2). Then substituting (2) back into (24) gives F_{n−1}/I_{n−1} = F_{−1} and hence, applying w^{−1}D_{w} to both sides, F_{n} = F, proving also part (i) of Theorem 2. Similarly, taking p = n − 2 in equations (20), (22), and (23) and then setting w = 0 gives

∑_{r=0}^{n−2}(n − r − 1) \frac{I′_{r}(x)}{I_{r}(x)} = \frac{n − 1}{2} \frac{n^{n} x}{1 − n^{n}x} ,

and integrating this and exponentiating gives (3).

Finally, we must prove the reflection symmetry (4). For this purpose, it is useful to construct the power series I_{p} in another way. Define a function F̃_{0} ∈ P by

F̃_{0}(w, x) = ∑_{d=0}^{∞} x^{d} \frac{∏_{r=1}^{r=nd}(nw+r)}{∏_{r=1}^{r=d}(w+r)^{n}}  (25)

and set F̃_{p}(w, x) = M^{p}F̃_{0}(w, x) for all p ≥ 0. Since F̃_{0}(w, x) is congruent to F(w, x) modulo w^{n}, we find by induction that F̃_{p}(w, x) is congruent to F_{p}(w, x) modulo w^{n−p} for all 0 ≤ p ≤ n − 1 and hence that I_{p}(x) = F̃_{p}(0, x) in this range. We now argue as above, using F̃_{0} instead of F_{−1}. This function satisfies the differential equation

(D_{w}^{n−1} − nx ∏_{j=1}^{n−1}(nD_{w} + j)) F̃_{0} = w^{n−1} .

Applying Corollary 1 repeatedly, we obtain

∑_{s=0}^{n−1−p} C̃_{s}^{(p)}(x) D_{w}^{s}F̃_{p}(w, x) = w^{n−p−1}

for 0 ≤ p ≤ n − 1, where the coefficients C̃_{s}^{(p)}(x) ∈ ℚ[[x]] can be calculated recursively, the top one being given by

C̃_{n−1−p}^{(p)}(x) = (1 − n^{n}x)I_{0}(x) ⋯ I_{p−1}(x).

Specializing to p = n − 1 and using (2), we find that F̃_{n−1}(w, x) = I_{n−1}(x) is independent of w. Now by downwards induction on p, using the equation F̃_{p} = I_{p}wD_{w}^{−1}F̃_{p+1}, we can “reconstruct” all of the power series F̃_{p}(w, x) (n − 1 ≥ p ≥ 0) from their special values I_{p}(x) = F̃_{p}(0, x) at w = 0, obtaining in particular the formula

w^{1−n} F̃_{0}(w, x) = I_{0} D_{w}^{−1} I_{1} D_{w}^{−1} ⋯ I_{n−2} D_{w}^{−1} I_{n−1}

for the initial series F̃_{0}. Comparing the coefficients of x^{d} on both sides of this equation, we find

\frac{n^{−1} ∏_{r=0}^{nd}(nw + r)}{[w(w + 1) ⋯ (w + d)]^{n}} = ∑_{{d_{0},...,d_{n−1}≥0 \atop d_{0}+···+d_{n−1}=d}} \frac{c_{0}(d_{0}) ⋯ c_{n−1}(d_{n−1})}{(w + d_{1} + ⋯ + d_{n−1})(w + d_{2} + ⋯ + d_{n−1}) ⋯ (w + d_{n−1})}

for all d ≥ 0, where c_{p}(d) denotes the coefficient of x^{d} in I_{p}(x). Splitting up the sum on the right into the subsum over n-tuples (d_{0}, …, d_{n−1}) with max{d_{r}} ≤ d − 1 and the sum over the n-tuples which are permutations of (d, 0, …, 0), and using that c_{p}(0) = 1 for all p, we can rewrite this equation as

∑_{p=0}^{n−1} \frac{c_{p}(d)}{w^{n−p−1}(w + d)^{p}} = \frac{∏_{r=0}^{nd}(nw + r)}{n ∏_{r=0}^{d}(w + r)^{n}} − ∑_{{0≤d_{0},...,d_{n−1}<d \atop d_{0}+···+d_{n−1}=d}} \frac{c_{0}(d_{0}) ⋯ c_{n−1}(d_{n−1})}{(w + d_{1} + ⋯ + d_{n−1}) ⋯ (w + d_{n−1})}

Now suppose by induction that c_{p}(d′) = c_{n−p−1}(d′) for all d′ < d and all 0 ≤ p ≤ n − 1. (Notice that this is true for d′ = 0 because c_{p}(0) = I_{p}(0) = 1 for all p, providing the starting point for the induction.) Then both terms on the right are (−1)^{n−1}-invariant under the map w → −w − d, as one sees for the second term by making the renumbering d_{r} → d_{n−1−r}. It follows that the left-hand side has the same invariance and hence that c_{p}(d) = c_{n−1−p}(d) for all 0 ≤ p ≤ n − 1, completing the inductive proof of the desired symmetry I_{n−1−p} = I_{p}.

2.3. Proof of Theorem 3

We now turn to the expansion of F(w, x) near w = ∞. We first prove Lemma 1, which said that any periodic fixed point of the map M : P → P has a logarithm which belongs to w ℚ[[x, w^{−1}]].

Proof of Lemma 1: The effect of M on logarithms is given by M(e^{H(w,x)}) = e^{H^{∗}(w,x)} , where

H^{∗}(w, x) = H(w, x) − H(0, x) + log(1 + \frac{DH(w, x) − DH(0, x)}{w});  (26)

here, as before, D denotes x\frac{∂}{∂x}. Suppose that H(w, x) := log F (w, x) is not O(w), and let e be the smallest integer such that the coefficient of x^{e} in H(w, x) is not O(w) as w → ∞. Then

H(w, x) = Cx^{e}w^{N} + xO_{w}(w) + x^{e}O_{w}(w^{N−1}) + O(x^{e+1})  (w → ∞)  (27)

for some C ≠ 0 and N ≥ 2, where O_{w}(w^{ν}) denotes a polynomial in x with coefficients that grow at most like w^{ν} as w → ∞ and O(x^{e+1}) denotes an element of x^{e+1} ℚ(w)[[x]]. From (26) and (27),

H^{∗}(w, x) = H(w, x) + Cex^{e}w^{N−1} + xO_{w}(1) + x^{e}O_{w}(w^{N−2}) + O(x^{e+1}).

This has the same form as (27) with the same C, e, and N. Iterating, we find that

log (M^{k}F (w, x)) = H(w, x) + kCex^{e}w^{N−1} + xO_{w}(1) + x^{e}O_{w}(w^{N−2}) + O(x^{e+1}),

and this contradicts the assumption that M^{k}F = F, since C ≠ 0 and N ≥ 2.

As already mentioned in the introduction, Lemma 1 implies that F(w, x) has an asymptotic expansion of the form (5). From the proof of the lemma, we see that each F_{p}(w, x) = M^{p}F(w, x) has an asymptotic expansion

F_{p}(w, x) ∼ e^{μ(x)w} ∑_{s=0}^{∞} Φ_{p,s}(x) w^{−s}  (w → ∞)  (28)

of the same form, with the same function μ(x) in the exponent. The equation F_{p+1} =MF_{p} gives

Φ_{0,s} = Φ_{s}, Φ_{p+1,s} = \frac{1 + μ′}{I_{p}} Φ_{p,s} + \begin{cases}(Φ_{p,s−1}/I_{p})′ & if s ≥ 1, \\ 0 & otherwise, \end{cases}  (29)

where f′ denotes Df = x df/dx. We want to solve these equations by induction on p for small s.

Before doing this, we begin with the following observation. Let L(x) = (1 − n^{n}x)^{−1/n} as in Theorem 3. Then (2) says that the product of the functions I_{p}(x)/L(x) (p ∈ ℤ/nℤ) equals 1, so if we define

H_{p}(x) = \frac{L(x)^{p}}{I_{0}(x) ⋯ I_{p−1}(x)}  (p ≥ 0),  (30)

then we have the properties

H_{0} = 1, H_{p}/H_{p+1} = I_{p}/L, H_{1}H_{2} ⋯ H_{n} = 1, H_{p+n} = H_{p}, H_{n−p} = H_{p}^{−1},  (31)

where the last equality is originally true for 0 ≤ p ≤ n but then, in view of the periodicity of {H_{p}}, holds for any p ∈ ℤ/nℤ. A number of identities below are simpler to state in terms of the functions H_{p}(x) than in terms of the original functions I_{p}(x).

The case s = 0 of (29) gives by induction the formula Φ_{p,0} = (1 + μ′)^{p}/I_{0} ⋯ I_{p−1}. Combining this with the formulas F_{n} = F and (2), we obtain (1 + μ′)^{n} = L^{n}, from which the first equation in (6) follows since μ(x) is a power series in x with no constant term. This also gives us the formula

Φ_{p,0}(x) = H_{p}(x) Φ_{0}(x) for all p ≥ 0,

with H_{p} as in (30). Now substituting this into the case s = 1 of (29) we find inductively

Φ_{p,1}(x) = H_{p}(x) ( Φ_{1}(x) + p \frac{Φ′_{0} − L′}{L} + \frac{Φ_{0}}{L} ∑_{r=1}^{p} \frac{H′_{r}}{H_{r}} ) for all p ≥ 0.

Setting p = n in this relation and using the third and fourth of equations (31) and F_{n} = F, we deduce that Φ_{0} = L, which is the second assertion of Theorem 3. At the same time we can refine the last two equations to

Φ_{p,0} = H_{p} L,  Φ_{p,1} = H_{p} (Φ_{1} + ∑_{r=1}^{p} \frac{H′_{r}}{H_{r}} )  (p ≥ 0).  (32)

The proof of the third identity in (6) is similar, but the calculations are more complicated. The case s = 2 of (29) gives by induction the formula

Φ_{p,2} = H_{p} (Φ_{2} + p (\frac{Φ_{1}}{L} )′ + (∑_{r=1}^{p} \frac{H′_{r}}{H_{r}} )\frac{Φ_{1}}{L} + \frac{1}{L} ∑_{s=2}^{p} ∑_{r=1}^{s−1} \frac{H′_{r}}{H_{r}} \frac{H′_{s}}{H_{s}} + (\frac{1}{L} ∑_{r=1}^{p−1}(p−r)\frac{H′_{r}}{H_{r}} )′)

for all p ≥ 0. Taking p = n, observing that

∑_{s=2}^{n} ∑_{r=1}^{s−1} \frac{H′_{r}}{H_{r}} \frac{H′_{s}}{H_{s}} ≡ \frac{1}{2}(( ∑_{p=1}^{n} \frac{H′_{p}}{H_{p}} )^{2} − ∑_{p=1}^{n} (\frac{H′_{p}}{H_{p}} )^{2}) = −\frac{1}{2} ∑_{p=1}^{n} (\frac{H′_{p}}{H_{p}} )^{2}

by the third equation in (31), and using F_{n} = F, we find that

n(\frac{Φ_{1}}{L} )′ = \frac{1}{2L} ∑_{p=1}^{n} (\frac{H′_{p}}{H_{p}} )^{2} + (\frac{1}{L} ∑_{p=0}^{n−1} p \frac{H′_{p}}{H_{p}} )′ = −\frac{(n + 1)(n − 2)}{24} (L^{n−1})′ ,

the last equation being Lemma 3 below. Integrating and using Φ_{1}(0) = 0 gives the last identity in (6).

Lemma 3. The functions {H_{p}(x)}_{p∈ℤ/nℤ} satisfy

\frac{1}{2L} ∑_{p (mod n)}(\frac{H′_{p}}{H_{p}} )^{2} = −(\frac{(n + 1)(n − 2)}{24} L^{n−1} + \frac{1}{L} ∑_{p=0}^{n−1} p \frac{H′_{p}}{H_{p}} )′ .  (33)

The proof consists of expressing the left-hand side of (33) in terms of the functions I_{0}, I_{1}, …, I_{n−1} and their derivatives, getting rid of all square terms via the product rule, and then eliminating I_{n−1}, I_{n−2}, and I_{n−3}. The last elimination is achieved by computing the coefficients C_{p}^{(n−3−p)} inductively by (20), starting with

C_{n−3}^{(0)} = −n^{n−2} S_{2}(n − 1) x = −\frac{(n − 1)(n − 2)(3n − 1)}{24}L′/L^{n+1} ,

and then setting p = n − 3, exactly as we did with C_{p}^{(n−1−p)} and C_{p}^{(n−2−p)} in Subsection 2.2 to prove eqs. (2) and (3). At this stage, all terms involving products of two functions I_{p} cancel, and the resulting expression can be integrated. We omit the details, which are somewhat tedious, since the last identity in (6) also follows easily from Theorem 4.

2.4. Proof of Theorem 4

We set X = L^{n} and Y = (L^{n} − 1)/n. Note that

D(μ) = L − 1,  D(L) = LY,  D(X) = X^{2} − X,  D(Y) = XY.  (34)

The first identity implies that D_{w} e^{μw} = e^{μw} D̃_{w}, where D̃_{w} = D + Lw. By induction on k, the powers of the differential operator D̃_{w} are given by

\begin{aligned}D̃_{w}^{k} & = ∑_{m=0}^{k} \binom{k}{m} D̃_{w}^{m}(1) D^{k−m} \\ & = D^{k} + k Lw D^{k−1} + \frac{k(k − 1)}{2} ((Lw)^{2} + Y (Lw)) D^{k−2} + …. \end{aligned}  (35)

A second induction gives the formula

D̃_{w}^{m}(1) = ∑_{j=0}^{m} H_{m,j}(X) (Lw)^{m−j},  (36)

with H_{m,j} ∈ ℤ[X, Y] ⊆ ℚ[X] given by (7).

The function F(w, x) satisfies the ODE

(D_{w}^{n} − w^{n} − x ∏_{j=1}^{n}(nD_{w} + j))F = 0 .

Since D_{w} e^{μw} = e^{μw} D̃_{w}, the function F̃(w, x) = e^{−μ(x)w}F(w, x) satisfies the differential equation 𝔏F̃ = 0, where 𝔏 is the differential operator

\begin{aligned}𝔏 & = L^{n} (D̃_{w}^{n} − w^{n} − x ∏_{j=1}^{n}(nD̃_{w} + j)) \\ & = D̃_{w}^{n} − (Lw)^{n} − (L^{n} − 1) ∑_{r=1}^{n} \frac{S_{r}(n)}{n^{r}} D̃_{w}^{n−r} . \end{aligned}

Using (35) and (36), we can expand 𝔏 as 𝔏 = ∑_{k=1}^{n}(Lw)^{n−k}𝔏_{k}, with 𝔏_{k} defined by (9). Combining the differential equation 𝔏F̃ = 0 with the asymptotic expansion F̃(w, x) ∼ ∑_{s≥0} Φ_{s}(x)w^{−s} for large w, we obtain (12).

We will next use (12) to prove by induction Φ_{s} belongs to Lℚ[L]. Since 𝔏_{1}(Lℚ[L]) = L^{2}Y ℚ[L], it suffices to show that

𝔏_{k}(Lℚ[L]) ⊆ L^{k+1}Y ℚ[L]  (2 ≤ k ≤ n).  (37)

Let I ⊂ ℚ[L] be the ideal generated by XY. Since D and Y commute modulo I by (34) and (D − rY)L^{r} = 0, we have

(D − Y)(D − 2Y) … (D − kY)L^{r} ∈ \begin{cases}L^{r} I & if 1 ≤ r ≤ k , \\ L^{r}Y ℚ[L] & if r ≥ k + 1 . \end{cases}

Therefore (37) is a consequence of the following lemma.

Lemma 4. For all k > 1,

𝔏_{k} ≡ \binom{n}{k}(D − Y)(D − 2Y) ⋯ (D − kY) (mod I) .

Proof: The recursion (7) for H_{m,j} shows that H_{m,j} ≡ h_{m,j} Y^{j} (mod I), where h_{m,j} ∈ ℤ is given recursively by

h_{0,j} = δ_{0,j},  h_{m,j} = h_{m−1,j} + (m − j)h_{m−1,j−1} ∀ m ≥ 1 .  (38)

Thus h_{m,j} = 𝔖_{m}^{(m−j)}, where 𝔖_{m}^{(k)} denotes a Stirling number of the second kind (the number of ways of partitioning a set of m elements into k non-empty subsets). We also note (1 − L^{n})n^{−r} ≡ (−1)^{r}Y^{r} (mod I) for all r ≥ 1. Combining these facts with (9), we find that

𝔏_{k} ≡ ∑_{i=0}^{k}( ∑_{r=0}^{k−i}(−1)^{r}\binom{n − r}{i} S_{r}(n) 𝔖_{n−r−i}^{(n−k)}) Y^{k−i}D^{i} (mod I) .

The desired congruence for 𝔏_{k} now follows from the generating series calculation

∑_{i=0}^{k}( ∑_{r=0}^{k−i}(−1)^{r}\binom{n − r}{i} S_{r}(n) 𝔖_{n−r−i}^{(n−k)}) t^{i}

= ∑_{i=0}^{n}( ∑_{r=0}^{n−i}(−1)^{r}\binom{n − r}{i} S_{r}(n) [\frac{1}{(n − k)!} ∑_{j=0}^{n−k}(−1)^{n−k−j}\binom{n − k}{j}j^{n−r−i}]) t^{i}

= \frac{1}{(n − k)!} ∑_{j=0}^{n−k}(−1)^{n−k−j}\binom{n − k}{j} ∑_{r=0}^{n}(−1)^{r}S_{r}(n) ∑_{i=0}^{n−r} \binom{n − r}{i}j^{n−r−i}t^{i}

= \frac{1}{(n − k)!} ∑_{j=0}^{n−k}(−1)^{n−k−j}\binom{n − k}{j} ∑_{r=0}^{n}(−1)^{r}S_{r}(n)(j + t)^{n−r}

= \frac{n!}{(n − k)!} ∑_{j=0}^{n−k}(−1)^{n−k−j}\binom{n − k}{j}\binom{j + t − 1}{n}

= \frac{n!}{(n − k)!}\binom{t − 1}{k} = \binom{n}{k} (t − 1)(t − 2) ⋯ (t − k) ,

where the first equality follows from the well-known fact that the expression in square brackets equals 𝔖_{n−r−i}^{(n−k)} if i + r ≤ k and 0 for i + r > k and the second-to-last equality is obtained by expanding (1 + u)^{t−1}((1 + u) − 1)^{t−1} by the binomial theorem and equating coefficients of t^{n}.

3. Further discussion of the large w expansion of F(w, x)

In this final section we give some further information and conjectures about the power series Φ_{s}(x) defined by equation (5). We begin by giving the numerical values for n ≤ 5 and s ≤ 4. For this purpose it is convenient to divide Φ_{s}/L by ((n − 2)(n + 1)/24n)^{s}/s! and write the result as the sum of (1 − L^{n−1})^{s} and a correction term, because the coefficients then become much simpler than without this renormalization:

n = 3 : s = 1 : 1 − L^{2}
s = 2 : (1 − L^{2})^{2}
s = 3 : (1 − L^{2})^{3} + 144 (1 − 5L^{3} + 4L^{6})
s = 4 : (1 − L^{2})^{4} + 576 (1 − 94L^{2} − 5L^{3} + 245L^{5} + 4L^{6} − 151L^{8})
n = 4 : s = 1 : 1 − L^{3}
s = 2 : (1 − L^{3})^{2}
s = 3 : (1 − L^{3})^{3} + \frac{36}{25} (4 + 72L − 297L^{5} + 221L^{9})
n = 4 (1 − L^{3})^{4} + \frac{144}{125} (884 + 360L − 20L^{3} − 19584L^{4} − 1485L^{5}
+44253L^{8} + 1105L^{9} − 25513L^{12})
n = 5 : s = 1 : 1 − L^{4}
s = 2 (1 − L^{4})^{2}
s = 3 : (1 − L^{4})^{3} + \frac{32}{45} (7 + 134L^{2} − 504L^{7} + 363L^{12})
s = 4 : (1 − L^{4})^{4} + \frac{16}{135} (168 + 8576L + 3216L^{2} − 168L^{4} − 127568L^{6}
−12096L^{7} + 270144L^{11} + 8712L^{12} − 150984L^{16})
Table: List of values of s! (\frac{24n}{(n−2)(n+1)})^{s}Φ_{s}/L for s = 1, 2, 3, 4 and n = 3, 4, 5

This suggests that the series ∑_{s}(Φ_{s}/L) w^{−s} is given to a first approximation by a pure exponential exp(\frac{(n−2)(n+1)}{24n}(1 − L^{n−1})/w) and hence that the formulas for the coefficients of the expansion (5) may become simpler if we take the logarithm. Doing this, we find an expansion which begins

log F(w, x) = μ(x)w + log L(x) + \frac{(n − 2)(n + 1)(1 − L(x)^{n−1})}{24n} w^{−1} + 0 w^{−2} + ⋯

and in which, at least experimentally, the coefficient of w^{−j} for j ≥ 0 is the sum of a term independent of x and a term of the form L^{−j} times a polynomial (without constant term) in L^{n}. This can be stated more elegantly by applying the operator w^{−1}D and adding 1, in which case it takes the form

1 + \frac{x}{w} \frac{∂}{∂x} log F(w, x) =^{?} L ∑_{k=0}^{∞} \frac{P_{k}(n, L^{n})}{(nLw)^{k}}  (39)

where P_{k}(n, X) is a polynomial in X of degree k with coefficients in ℚ[n], the first values being

P_{0}(n, X) = 1 ,
P_{1}(n, X) = X − 1 ,
P_{2}(n, X) = − \frac{(n + 1)(n − 1)(n − 2)}{24} (X − 1)X ,
P_{3}(n, X) = 0 ,
P_{4}(n, X) = \frac{(n + 1)(n − 1)(n − 2)}{5760} (X − 1)(A_{3}X^{3} + A_{2}X^{2} + A_{1}X) ,
P_{5}(n, X) = − \frac{(n + 1)(n − 1)(n − 2)}{5760} (X − 1)(B_{4}X^{4} + B_{3}X^{3} + B_{2}X^{2} + B_{1}X)

with

A_{1} = (n − 3)(7n^{3} − 17n^{2} + 22n − 24) ,
A_{2} = − (2n − 3)(3n − 1)(7n^{2} − 9n + 14) ,
A_{3} = 3 (14n^{4} − 33n^{3} + 52n^{2} − 23n − 2) ,
B_{1} = − (n − 3)(n − 4)(7n^{3} − 17n^{2} + 22n − 24) ,
B_{2} = 2 (n − 1)(n − 2)(49n^{3} − 115n^{2} + 152n − 124) ,
B_{3} = −4 (n − 1)(3n − 1)(3n − 4)(7n^{2} − 9n + 14) ,
B_{4} = 8 (n − 1)(3n − 2)(7n^{3} − 11n^{2} + 17n − 1) .

The coefficients of the polynomials P_{k} follow no apparent pattern apart from the divisibility by (n + 1)(n − 1)(n − 2)X(X − 1): the common factors of A_{1} and B_{1} and of A_{2} and B_{3} are striking, but nothing similar occurs for the next two polynomials. On the other hand, there is a simple formula for the leading coefficient of P_{k}(n, X) with respect to n, namely (at least up to k = 7)

P_{k}(n, X) = \begin{cases}α_{j} e_{k}(X) n^{4j−1} + O(n^{4j−2}) & if k = 2j > 0; \\ (j − 1)α_{j} e_{k}(X) n^{4j} + O(n^{4j−1}) & if k = 2j + 1, \end{cases}

where α_{j} denotes the coefficient of u^{2j} in \frac{u/2}{sinh u/2} (α_{0} = 1, α_{1} = −\frac{1}{24}, α_{2} = \frac{7}{5760}, α_{3} = −\frac{31}{967680}, …) and where e_{1} = X − 1, e_{2} = X^{2} − X, e_{3} = 2X^{3} − 3X^{2} + X, … are the polynomials defined by

e_{k}(X) = ∑_{l=1}^{k} (−1)^{k−l} (l − 1)! 𝔖_{k}^{(l)} X^{l} ∈ ℤ[X]

with 𝔖_{k}^{(l)} as before a Stirling number of the second kind. This is interesting because the argument X = L^{n} of P_{k}(n, X) in equation (39) is in fact (1 − n^{n}x)^{−1} and the functions e_{k}((1 − x)^{−1}) have the basic property

e_{k}(\frac{1}{1 − x}) = ∑_{d=1}^{∞} d^{k−1}x^{d} ∈ x ℤ[[x]] (k ≥ 1) .

There is also a possible intriguing connection with modular and elliptic functions since, for example, the power series in two variables ∑ α_{j}e_{2j}(\frac{1}{1−x}) u^{2j−1} is closely related to the expansion of the Weierstrass ℘-function and related Jacobi forms.

As a final remark, we observe that (39), if it is true, defines the power series F(w, x) even for non-integral values of n and shows that this function is analytic in n as well as in w and x. This seems surprising since F is defined as a hypergeometric function of order n and we would usually not expect such series to have an interpolation with respect to the order of the differential equation which they satisfy.

References

[BCOV] M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Holomorphic Anomalies in Topological Field Theories, Nucl. Phys. B405 (1993), 279–304.

[CaDGP] P. Candelas, X. de la Ossa, P. Green, L. Parkes, A Pair of Calabi-Yau Manifolds as an Exactly Soluble Superconformal Theory, Nuclear Phys. B359 (1991), 21–74.

[Z1] A. Zinger, The Reduced Genus-One Gromov-Witten Invariants of Calabi-Yau Hypersurfaces, math/0705.2397.

[Z2] A. Zinger, Standard vs. Reduced Genus-One Gromov-Witten Invariants, math/0706.0715.

Max-Planck-Institut für Mathematik, Bonn zagier@mpim-bonn.mpg.de

Department of Mathematics, SUNY Stony Brook, NY 11794-3651 azinger@math.sunysb.edu