%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Volume recursion via $q$-brackets}
\label{sec:D2rec}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we define recursively polynomials in the ring 
$R = \QQ[h_1,h_2,\ldots]$ and show that they compute volumes of the strata 
after a suitable specialization. The method of proof relies on lifting 
the $E_2$-derivative via the Bloch-Okounkov $q$-bracket and expressing 
cumulants in terms of this lift. This recursion looks quite different from 
the recursion given in Theorem~\ref{intro:VolRec}, since it is only defined 
on the level of polynomials in the variables $h_i$ and requires
$h_i$-derivatives. 
\par
To define the substitution, first recall the numbers~$b_j$ introduced
in~\eqref{eq:defBandb}. We let
\be \label{eq:defalpha}
P_B(u) \= \exp\Bigl(- \sum_{ j \geq 1} j!\, b_{j+1} u^{j+1}\Bigr)
\quad \text{and}  \quad
\alpha_\ell \= \bigl[u^{\ell}\bigr] \frac{1}{(u/P_B(u))^{-1}}\,,
\ee
where the denominator denotes the inverse function of $u/P_B(u)$.
For the recursion we define for a finite set  $I=\{i_1,\ldots,i_n\}$
of positive 
integers the formal series $\mathcal{H}_I\in R[[z_{i_1}, \ldots, z_{i_n}]]$
if $|I| \geq 2$ and $\mathcal{H}_{\{i\}} \in \frac{1}{z_i} R[[z_{i}]]$ by
\begin{flalign}
&\mathcal{H}_{\{i\}} \= \frac{1}{z_i} + \sum_{\ell\geq 1} h_\ell z_i^\ell\,, \nonumber \\ 
&\cHH_{\{i,j\}} \= \frac{z_i \cHH'(z_i) - z_j\cHH'(z_j)}
{\cHH(z_j) - \cHH(z_i)} -1 \nonumber  \\
&\,\, \= 2h_1 z_iz_j + h_2 (3z_i^2 z_j  + 3z_i z_j^2) + 
4h_3 z_i^3z_j  + (2h_1^2 + 4h_3 )z_i^2z_j^2 + 4h_3 z_iz_j^3  + \cdots\,, \nonumber \\
&\mathcal{H}_{I} = \frac{1}{2(n-1)}\sum_{I=I'\sqcup I'' \atop I', I'' \neq \emptyset}
 D_2(\mathcal{H}_{I'},\mathcal{H}_{I''})\,, \label{eq:Hindform}
\end{flalign}
where we abbreviate $\mathcal{H}_{n}=\mathcal{H}_{[\![1,n]\!]}$, $\cHH = \cHH_1$
and  $h_{\ell_1,\ldots,\ell_n} = [z_1^{\ell_1}\dots z_n^{\ell_n}]\mathcal{H}_{n}$
and where the symmetric bi-differential operator $D_2$ is defined by
\be \label{eq:D2H}
D_2(f,g) \= \sum_{\ell_1,\ell_2 \geq 1}
h_{\ell_1,\ell_2} \,
\frac{\partial f}{\partial h_{\ell_1}}\,  \frac{\partial g}{\partial h_{\ell_2}} \,.
\ee
\par
\begin{Thm} \label{thm:D2recursion}
The rescaled volume of the stratum with signature $\mu=(m_1,\ldots,m_n)$ can be computed as 
\bes
v(\mu) \=  \frac{(2\pi i)^{2g}}{(2g-2+n)! } \, \bigl.
h_{m_1+1,\ldots,m_n+1}\bigr|_{h_\ell \mapsto \alpha_\ell} 
\ees
using the recursion~\eqref{eq:Hindform} and the values of the $\alpha_\ell$ in~\eqref{eq:defalpha}.
\end{Thm}
\par

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Three sets of generators for the algebra of 
shifted symmetric functions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We let $\Lambda^*$ be the algebra of shifted symmetric functions
(see e.g.\ \cite{eo}, \cite{zagBO} or \cite{cmz}) and recall the standard 
generators  
\be \label{eq:defpk}
p_\ell(\l) \= \sum_{i=1}^\infty \left( (\l_i -i +\h)^\ell - (-i + \h)^\ell \right)
\+ (1-2^{-\ell})\,\zeta(-\ell) %\= \ell!\,Q_{\ell+1}(\lambda)
\,. 
\ee
Note that $(1-2^{-\ell})\,\zeta(-\ell)\= \ell! b_{\ell+1}$. The algebra $\Lambda^*$ 
is provided with a grading where each~$p_\ell$ has weight $\ell+1$. For Hurwitz 
numbers the geometrically interesting generators are
\be \label{eq:defssf}
f_{\ell}(\lambda) \= z_{\ell} \chi^\lambda(\ell)/\dim \chi^\lambda\,,
\ee
where $z_{\ell}$ is the size of the conjugacy class of the cycle
of length~$\ell$, completed by singletons. The first few of these functions 
are
\bes f_1 \= p_{1} + \frac{1}{24}, \quad f_2   \=  \frac{1}{2} p_{2},
\quad f_3  \= \frac{1}{3} p_{3} - \frac{1}{2} p_{1}^2
 + \frac{3}{8} p_{1} + \frac{9}{640}\,.
\ees 
The third set of generators, defined implicitly by Eskin and Okounkov, will 
serve as top term approximations of $f_\ell$. We define $h_\ell \in \Lambda^*$
by
\be \label{eq:hldef} 
h_\ell \= \frac{-1}{\ell}  [u^{\ell+1}] P(u)^\ell 
\quad \text{where} \quad
P(u) \= \exp \Bigl( -\sum_{s \geq 1} u^{s+1} p_s \Bigr)\,.
\ee
Observe that by definition~$h_\ell$ has pure weight~$\ell+1$. The first few of 
these functions  are
\bes h_1  \= p_{1}, \quad h_2   \= p_{2},
\quad h_3  \=  p_{3} - \frac{3}{2} p_{1}^2 \,.
\ees
\par 
\begin{Prop}[{\cite[Theorem~5.5]{eo}}] \label{prop:EOconv}
The difference $f_\ell -  h_\ell/\ell$ has weight strictly 
less than~$\ell+1$.
\end{Prop}
\par
We abuse the notation $h_\ell$ for generators of $R$ and for elements 
in $\Lambda^*$. This is intentional and should not lead to confusion, since
the map $h_\ell \mapsto h_\ell$ induces an isomorphism of algebras $R \cong \Lambda^*$, 
by the preceding proposition.
\par
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The lift of the evaluation map to the Bloch-Okounkov ring}
\label{sec:lift}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $f\colon\P\to\QQ$ be an arbitrary function on the set $\P$ of all partitions.
Bloch and Okounkov (\cite{blochokounkov}) associated to~$f$ the formal power series
\be \label{defqbrac} 
  \sbq f \= \frac{\sum_{\l\in\P} f(\l)\,q^{|\l|}}{\sum_{\l\in\P} q^{|\l|}}\;\,\in\;\QQ[[q]]\,,
\ee
which we call the {\em $q$-bracket}, and proved that this $q$-bracket is 
a quasimodular form of weight~$k$ whenever~$f$ belongs to the subspace 
of ~$\Lambda^*$ of weight~$k$ (see \cite{blochokounkov}, and \cite{zagBO} 
or \cite{GoujM} for alternative proofs).  
\par
In \cite[Section~8]{cmz} we studied in detail an evaluation map $\evX$
(implicitly defined in \cite{eo}) on the ring of quasimodular forms that
measures the growth rate of the coefficients of quasimodular forms,
or equivalently, their asymptotics as $\tau \to 0$ along the
imaginary axis (\cite[Proposition~9.3]{cmz}). The purpose of this section is
to lift this evaluation map to the Bloch-Okounkov ring and to express it in
terms of the generators~$h_i$ introduced in the previous section.
\par
The map $\evX$ is the algebra homomorphism from the ring of quasimodular forms
$\wM_* = \QQ[E_2,E_4,E_6]$ to~$\QQ[X]$, sending the 
Eisenstein series $E_2$ (normalized to have constant coefficient one) 
to~$X+12$, $E_4$ to~$X^2$, and $E_6$ to~$X^3$.
In this way, the larger the degree of $\evX(f)$, the larger the (polynomial)
growth of the coefficients of~$f$, see \cite[Proposition~9.4]{cmz} for the
precise statement. It is also convenient to work with the evaluation
map\footnote{This would be $(2\pi i)^k \ev[F](-4\pi^2 \hslash)$ in
the notation of \cite[Equation~(85)]{cmz}.}
\be\label{defevh} \evh[F](\hslash) \= \frac1{\hslash^{k/2}}\,
\evX[F] \Bigr|_{X \mapsto \frac{1}\hslash}
\;\in\;\QQ[1/\hslash]\,\quad \text{for} \quad F \in \wM_k\,. \ee
We also use  the brackets $\sbrX f:=\evX\bigl[\sbq f\bigr](X)$ and
$\sbrh f:=\evh\bigl[\sbq f\bigr](\hslash)$ for $f\in\Lambda^*$ as abbreviation. 
Note that~$\Lambda^*$ admits a natural ring homomorphism to~$\Q$, 
the evaluation at the emptyset, explicitly given by the 
map $p_\ell \mapsto \ell! b_{\ell+1}$.
\par
\begin{Prop} \label{prop:liftviadelta}
  There is a second order differential operator $\Delta\colon \Lambda^* 
\to \Lambda^*$ of degree~$-2$ and a derivation $\partial\colon\Lambda^* 
\to \Lambda^*$ of degree~$-1$ such that
\be
\brh f \= \frac{1}{\hslash^k}(e^{\hslash(\Delta - \p^2 - \p/\p p_1)/2} f)\,(\emptyset)\,
\ee
for $f \in \Lambda^*_k$ homogeneous of weight~$k$. 
\end{Prop}
The differential operators are given in terms of the 
generators~$p_\ell$ by
\be
\partial(f) \= \sum_{i \geq 2} ip_{i-1} \frac{\partial}{\partial p_i}
\quad \text{and}\quad
\Delta(f) \= \sum_{k,\,\ell\,\ge\,1} (k+\ell) \,p_{k+\ell-1}\,\frac{\p^2}{\p p_{k}\,\p p_{\ell}} \,.
\ee
%sum(k=1,N,sum(l=1,N,(k+l)*plong[k+l-1]*dp(dp(f,k),l)));
\par
\begin{proof}  From the definition and \cite[Proposition~9.2]{cmz}
we deduce that the evaluation map can be computed for any $F\in\wM_k$ as
\be \label{eq:defevh}
\evh[F](\hslash) \= \frac1{\hslash^{k}}a_0\bigl(e^{\hslash\fd}  F\bigr) 
\ee
where $\fd  = 12 \partial/\partial E_2$ and where $a_0\colon F\mapsto F(\infty)$ 
is the constant term map from $\wM_*$ to~$\Q$. From \cite[Proposition~8.3]{cmz}
we deduce (note that differentiation with respect to~$Q_i$ in loc.\ cit.\
gives the extra $p_1$-derivative here) that the differential operators
defined above have the property that 
\be\label{E2deriv}  \fd\,\sbq f \= \bq{\h\bigl(\Delta\,-\,\pp^2
-  \p/\p p_1\bigr)\, f} \qquad (f \in \Lambda^*)\,.\ee
Since the constant term of the $q$-bracket of $f$ is in $\Lambda^*$, the
claim follows from these two equations.
\end{proof}
\par
To motivate the next section, we recall the notion of cumulants. 
Let $R$ and $R'$ be two commutative $\Q$-algebras with unit and $\la\;\,\ra \colon R\to R'$
a linear map sending~1 to~1.  (Of course the cases of interest to us will be when
$R$ is the Bloch-Okounkov ring~$\Lambda^*$ and $\la\;\,\ra$ is the $q$-, $X$-, or 
$\hslash$-bracket to $R'=\wM_*$, $\Q[X]$, or $\Q[\pi^2][\hslash]$,
respectively.) Then we extend
 $\la\;\,\ra$ to a  multi-linear map $R^{\otimes n}\to R'$ for every $n\ge1$, 
called {\em connected brackets}, 
the image of $g_1\otimes\cdots\otimes g_n$  being denoted by 
$\la g_1|\cdots|g_n\ra$,  
%or $\la|g_1\otimes\cdots\otimes g_n|\ra$,
that we define by 
\be  \label{slash}
\langle g_1|\cdots|g_n\rangle \= \sum_{\alpha \in \PPP(n)} (-1)^{\ell(\a)-1} 
(\ell(\a)-1)!\, \prod_{A\in\a} \Bigl\la \prod_{a\in A} g_a \Bigr\ra \,.
\ee
The most important property of connected brackets is their appearance 
in the logarithm of the original bracket applied to an exponential:
\bas \log\bigl(\bigl\la e^{g_1+g_2+g_3+\cdots}\bigr\ra\bigr)
   &\= \log\Bigl(1 \+ \sum_i\la g_i\ra \+ \frac1{2!}\sum_{i,j}\la g_ig_j\ra
     \+ \frac1{3!}\sum_{i,j,k}\la g_ig_jg_k\ra \+\cdots\Bigr) \\
  &\= \sum_i\la g_i\ra \+ \frac1{2!}\sum_{i,j}\la g_i|g_j\ra 
   \+ \frac1{3!}\sum_{i,j,k}\la g_i|g_j|g_k\ra \+\cdots \;. \eas
\par
We specialize to the Bloch-Okounkov ring~$\Lambda^*$
and we want to compute the leading terms of the connected brackets
associated with the $\la\cdot\ra_X$- or $\la\cdot\ra_{\hslash}$-brackets. 
Recall from \cite[Proposition~11.1]{cmz}:
\par
\begin{Prop} \label{prop:degdrop}
Let $g_i \in \Lambda^*_{\leq k_i}$ ($i=1,\ldots,n$) be elements
of weight less than or equal to~$k_i$ and let $g_i^\top \in \Lambda^*_{k_i}$
be their top weight components. Let $k=k_1+\cdots+k_n$ be the total weight.
Then $\deg(\la g_1|\cdots|g_n\ra_X) \leq 1-n+k/2$ and 
\be \label{eq:ggtop}
 [X^{1-n +k/2}] \,\la g_1|\cdots|g_n\ra_X 
\=  [X^{1-n +k/2}] \,\la g_1^\top|\cdots|g_n^\top\ra_X\,.
\ee
\end{Prop}
The leading terms of the brackets are consequently
\ba \label{eq:deflead}
\la g_1|\cdots|g_n\ra_L &\,=\, [X^{1-n +k/2}] \,\la g_1|\cdots|g_n\ra_X 
\= \lim_{X\to\infty} \frac{\evX[\langle g_1|\cdots|g_n\rangle_q](X)}{X^{1-n+k/2}} \\
&\= [\hslash^{-k-1+n}] \,\la g_1|\cdots|g_n\ra_\hslash 
\= \lim_{\hslash\to 0} \hslash^{k+1-n}\,\evh[\langle g_1|\cdots|g_n\rangle_q](\hslash)\,.
\ea
We call them {\em rational cumulants}.
\par

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The cumulant recursion}
\label{sec:cumrec}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we prove a formula for computing the connected brackets
associated with the $\sbq\cdot$- or rather the $\sbrh\cdot$-brackets.
The core mechanism for their computation is 
summarized in the following purely algebraic property.
\par
Let $R$ be an $\NN$-graded commutative $\QQ$-algebra with $R_0 = \QQ$, 
complete with respect to the maximal ideal $\frakm = R_{>0}$. The following
statement gives a general recursion for expressions that appear in cumulants.
We will specialize~$R$ to the Bloch-Okounkov ring subsequently.
\par
\begin{KeyLe} \label{KL:D2recursion}
Suppose that $D\colon R \to R$ is a linear map. Then the following statements 
are equivalent: 
\begin{itemize}
\item[(1)] We have $D(x^3) - 3xD(x^2) + 3x^2D(x) = 0$ for all $x \in R$.
\item[(2)] For all $x \in R$ and all $n \geq 2$ 
$$D(x^n) = \binom{n}{2}D(x^2)x^{n-2} - n(n-2)D(x)x^{n-1} \,.$$
\item[(3)] For all $x,y,z \in R$
$$ D(xyz) \= xD(yz)+yD(xz)+zD(xy) - xyD(z)-xzD(y)-yzD(x)\,.$$
\item[(4)]  If we denote by $D_2\colon R^2 \to R$ the symmetric bilinear form
$$ D_2(x,y) = D(xy) - xD(y) -yD(x)$$
then 
$$ D(x_1\cdots x_n) \= \sum_{i=1}^n D(x_i) x_1\cdots \widehat{x_i}\cdots x_n
+ \sum_{1\leq i<j\leq n} D_2(x_i,x_j) x_1\cdots \widehat{x_i}\cdots \widehat{x_j}\cdots x_n $$
for all $x_1,\ldots,x_n \in R$.
\item[(5)] For any fixed~$x \in R$, the bilinear form  $D_2(x,y)$ is a
derivation in~$y$.
\item[(6)] The map $D \in {\rm Sym}^2({\rm Der}(R))$, i.e.\ $D$ is
a second order differential operator without constant term.
\item[(7)] For all $X \in \frakm$ there exists $\cLL(X) \in R$ such that
\be \label{eq:abstractcumu}
\log(e^{\hslash D}(e^{X/\hslash})) \= \frac{1}{\hslash} \cLL(X) \+ O(1)\quad(\hslash \to 0)\,.
\ee 
\end{itemize}
If any of these statements holds, the leading 
term of~\eqref{eq:abstractcumu} is given by $\cLL(X) = L(1)$, where
\be L(0)\= X\,, \quad L'(t)\= \frac12 D_2(L(t),L(t))\,.\ee
\end{KeyLe}
\par
\begin{proof}
The implication $(1) \Rightarrow (3)$ follows by passing to
the polarization. The implication $(3)\Rightarrow (4)$ can be proved by induction (replace
$x_1$ by $x_0 x_1$). The implications $(2) \Rightarrow (1)$ and 
$(4) \Rightarrow (1),(2),(3)$ follow by specialization.
The equivalence $(5) \Leftrightarrow (3)$ follows by direct computation.
To show $(6) \Leftrightarrow (5)$ think deeply.
To prove $(7) \Rightarrow (1)$ it suffices to consider the cubic term: 
the coefficient of $1/\hslash^2$ is the expression in~(1).
\par
To prove $(6) \Rightarrow (7)$ and the final formula for $\cLL(X)$ we
write
\be \label{eq:exphD}
e^{\hslash D}(e^x) \= e^{y(\hslash)} \,.
\ee
Then $y(0)=x$. Note that $(2)$ implies that  
$$ D(e^x) \= \frac12 D(x^2)e^x + D(x) (1-x)e^x\,.$$
Differentiation of~\eqref{eq:exphD} with respect to~$\hslash$ implies that $y' =D(y)+\tfrac12 D_2(y,y)$. Equivalently, writing
$y \= \sum_{n\geq 0} f_n(x) \hslash^n$, then the initial condition is that $f_0(x)=x$, and 
$$ (n+1)f_{n+1}(x) \= D(f_n) + \frac12 \sum_{m=0}^n D_2(f_m(x),f_{n-m}(x))\,.$$ 
Recursively this implies that $f_n(X/\hslash) = \cLL_n(X) \hslash^{-n-1} + O(\hslash^{-n})$ with
$\cLL_0(X)=X$ and 
$$  (n+1)\cLL_{n+1}(X) = \frac12 \sum_{m=0}^n D_2(\cLL_m(X),\cLL_{n-m}(X))\,.$$ 
We now let $L(t) =\sum_{n\geq 0} \cLL_n(X) t^n$ and the claims follow.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Application to volume computations}
\label{sec:appvol}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We now return to the proof of the main theorem of this section.
Recall the main idea from \cite{eo} that the volume of a stratum 
is given by the growth rate of the number of connected torus covers
and thus to the leading terms of cumulants of the~$f_\ell$. More
precisely for $2g-2 = \sum_{i=1}^n m_i$, the same argument as in \cite[Proposition~19.1]{cmz} 
gives that 
\be \label{eq:cumutovol}
{\rm vol}\,(\omoduli[g,n](m_1,\ldots,m_n)) \= (2\pi i)^{2g}  
\frac{\bL{f_{m_1+1}| \cdots |f_{m_n+1}}}{ (2g-2+n)! } \,.
\ee
\par
\begin{proof}[Proof of Theorem~\ref{thm:D2recursion}, one variable case]
First, $P_B(u) = P(u)|_{p_\ell \mapsto \ell! b_\ell} = P(u)(\emptyset)$.
Next, recall that Lagrange inversion for a power series $F \in u\CC[[u]]$
with non-zero linear term and inverse $G(z)$ states that
$k[z^k]G^n = n [u^{-n}]F^{-k}$ for $k,n \neq 0$. We apply this
to  $F = u/P_B(u)$ and to $k=2g-1$ and $n=-1$ to obtain that 
\ba \label{eq:fellinversion}
\frac{(2g-1)!}{(2\pi i)^{2g}} v(2g-2) &\= (2g-1)\bL{f_{2g-1}} \= \bL{h_{2g-1}} \\
&\= \frac{-1}{2g-1} [u] (P_B(u)/u)^{2g-1}
\= \bigl[u^{2g-1}\bigr] \frac{1}{(u/P_B)^{-1}}
\ea
using Proposition~\ref{prop:EOconv}, \eqref{eq:hldef} and Lagrange inversion.
\end{proof} 
\par
We pause for a moment to check the initial condition of the theorem
in the previous section independently of the Hermitian metric extension problem along the boundary of the strata. 
\par
\begin{proof}[Proof of~\eqref{eq:volminviaa} using~\eqref{eq:fellinversion}]
We want to show that $(2g-1)^2 \,a (2g-2) = \bL{h_{2g-1}}$.
Recall that a version of Lagrange inversion 
(see e.g.\ \cite[Formula~(2.2.8)]{Gessel}), in fact the case $k=0$
excluded in the version of the previous proof, states that
if $F \in z + z^2\CC[[z]]$ with composition inverse~$G(u)$, then for any Laurent
series $\phi(z)$
\be \label{eq:LagLogar}
[z^0] \phi(F)  \= [u^0] \phi(u) + [u^{-1}] \phi'(u) \log(G/u) \,.
\ee
If we let $\widetilde{\cAA}(z) =  1/z + \sum_{g\geq 1} \bL{h_{2g-1}} \, z^{2g-1}$, then we
need to show that $\widetilde{\cAA}(z) = \cAA(z)$.
We apply Lagrange inversion to $\phi(z) = z^{-2g}$ and $F = 1/\widetilde{\cAA}(z)$ to obtain that 
\bas
\frac{1}{2g!}[z^{0}]\widetilde{ \cAA}^{2g} &\= \frac{1}{2g!}[z^0] \phi(1/\widetilde{\cAA}(z)) 
\= \frac{-1}{(2g-1)!} [u^{2g}] \log(1/P_B) \\
&\=  \frac{-1}{(2g-1)!}  [u^{2g}]\,\sum_{s \geq 1} s!b_{s+1} u^{s+1}
\= [u^{2g}]B(u) = \frac{1}{(2g)!}[z^{0}] \cAA^{2g}
\eas
using~\eqref{eq:fellinversion} and~\eqref{eq:ind1}. This implies the claim.
\end{proof}
\par
For the general case of the theorem, we apply Section~\ref{sec:cumrec} to the
differential operator 
\be \label{eq:defD}
D= \tfrac12 (\Delta - \p^2 -  \p/\p p_1)\,.
\ee
\par
\begin{Prop} \label{prop:D2fg}
The bilinear  differential operator~$D_2$ defined in~\eqref{eq:D2H}
is the polari\-zation of $D$, namely, $D_2(f,g) = D(fg)-fD(g)-gD(f)$
for all~$f$ and~$g$.
\end{Prop} 
\par
\begin{proof} In terms of the $p_\ell$-generators the polarization is given by
\be
D_2(f,g) \= \sum_{k,\ell \geq 1} 
\Bigl((k+\ell)p_{k+\ell-1} - k\ell p_{k-1} p_{\ell-1} \Bigr)
\frac{\partial f}{\partial p_k}
\frac{\partial g}{\partial p_\ell} \,.
\ee
The definition~\eqref{eq:hldef} of~$h_\ell$ in terms of~$p_\ell$ implies that 
\be \label{eq:auxsum}
\frac{\partial \cHH(z)}{\partial p_k} \= -\frac{z \cHH'(z)}{\cHH(z)^{k+1}}
\quad \text{and} \quad
\sum_{n \geq 2} n p_{n-1} \cHH^{-n}(z) \= -\cHH(z)/z\cHH'(z)-1\,.
\ee
We compute that
\bas &\phantom{\=} \,\,  \sum_{k,\ell \geq 1} 
\Bigl((k+\ell)p_{k+\ell-1} - k\ell p_{k-1} p_{\ell-1} \Bigr) \frac{z_1 \cHH'(z_1)}
     {\cHH(z_1)^{k+1}} \cdot \frac{z_2 \cHH'(z_2)}{\cHH(z_2)^{\ell+1}} \\
&\= \frac{z_1 \cHH'(z_1)z_2\cHH'(z_2)}{\cHH(z_1)\cHH(z_2)} 
\Bigl( \sum_{n \geq 2} n p_{n-1} \frac{\cHH(z_1)^{1-n}-\cHH(z_2)^{1-n}}
{\cHH(z_2) - \cHH(z_1)} \\
& \phantom{\= \frac{z_1\cHH'(z_1)z_2\cHH'(z_2)}{\cHH(z_1)\cHH(z_2)}mm} -
\Bigl(1 + \frac{\cHH(z_1)}{z_1\cHH'(z_1)}\Bigr)\Bigl(1 + \frac{\cHH(z_2)}
{z_2\cHH'(z_2)}\Bigr) 
\Bigr) \\
&\= \cHH_2(z_1,z_2)\,,
\eas
and this implies the claim by the chain rule.
\end{proof}
\par
We now define the partition function of $h$-brackets  
\begin{flalign} \Phi^H(\bfu)_q & \= \Bigl\langle \exp\Bigl(\sum_{\ell \geq 1} h_\ell u_\ell\Bigr)\,\Bigr\rangle_q
  \=\sum_{\bfn\geq 0}\,\langle\underbrace{h_1\cdots h_1}_{n_1} \underbrace{h_2\cdots h_2}_{n_2}\cdots\rangle_q \,\frac{\bfu^\bfn}{\bfn!} 
   \nonumber  \\   \label{eq:defPhi}   
  &\=\sum_{n=0}^\infty \frac1{n!} \sum_{\ell_1,\ldots,\ell_n \geq 1} \langle h_{\ell_1} \cdots h_{\ell_n} \rangle_q \, u_{\ell_1}\cdots u_{\ell_n}\;
\end{flalign}
in the $h_\ell$-variables. Then the partition function of the
rational cumulants for the $h_\ell$-generators
\ba \label{eq:psiHq}
\Psi^H(\bfu)_q &\= \sum_{\bfn \geq 0}\la\underbrace{h_1|\cdots|h_1}_{n_1}|\underbrace{h_2|\cdots|h_2}_{n_2}|\cdots\ra_q\,\frac{\bfu^\bfn}{\bfn!}
  \= \log \Phi^H(\bfu) _q\,
\ea
is simply the logarithm of $\Phi^H$.
\par
\begin{proof}[Proof of Theorem~\ref{thm:D2recursion}, general case]
We first show that the pieces of $\Phi^H$ sorted by total degree in~$\bfu$
can be recursively computed using the $D_2$-operator. For this purpose
we let $\wt h_i = \hslash^{-i} h_i$. From the definition of cumulants, equation~\eqref{eq:defevh}
and $a_0(\la g \ra_q) = g(\emptyset)$, we obtain that 
\ba
\sum_{\bfn \geq 0}\la\underbrace{\wt h_1|\cdots|\wt h_1}_{n_1}|
\underbrace{\wt h_2|\cdots|\wt h_2}_{n_2}| \cdots\ra_{\hslash}\,\frac{\bfu^\bfn}
{\hslash^{|\bfn|}\, \bfn!}  
&\= \log\Bigl(\la \exp({\frac1\hslash} \sum_{i \geq 1} \wt h_i u_i)
\ra_\hslash\Bigr) \\
&\= \log\Bigl(e^{\hslash D} \exp\Bigl({\tfrac1\hslash} \sum_{i \geq 1} 
\wt h_i u_i\Bigr)\Bigr)(\emptyset)\,.
\ea
By applying the Key Lemma with $X =  \sum_{i \geq 1} \wt h_i u_i$ and undoing the
rescaling of the $h_i$ using~\eqref{eq:deflead} we obtain that 
\ba
\sum_{\bfn \geq 0}\la\underbrace{h_1|\cdots|h_1}_{n_1}|
\underbrace{h_2|\cdots|h_2}_{n_2}| \cdots\ra_L\,\frac{\bfu^\bfn}{ \bfn!}  
= \biggl(\sum_{n=0}^\infty \cLL_n\biggr) (\emptyset)
\ea
with
\be \label{eq:lrec}
\cLL_0 \= \sum_{n \geq 1} h_i u_i \quad \text{and} \quad
\cLL_n \= \frac1{2n} \sum_{r+s = n-1} D_2(\cLL_r,\cLL_s)
\ee
for $n>0$. Now define a linear map $\mho_n\colon \QQ[\bfu] \to \QQ[\bfz]$ by
  $$\mho_n(u_{\ell_1} \cdots u_{\ell_n}) \= {\rm Symm}
  (z_1^{\ell_1} \cdots z_n^{\ell_n})
  $$
and zero for monomials of length different from~$n$,
where ${\rm Symm}$ denotes symmetrization with respect to the $S_n$ action
on the variables~$z_i$. In this notation $\cHH_1 =  \mho_n \cLL_{n-1}$ and 
$$  \cHH_n \= \mho_n \Bigl( 
\sum_{\bfn \geq 0}\la\underbrace{h_1|\cdots|h_1}_{n_1}|\underbrace{h_2|\cdots|h_2}_{n_2}|
\cdots\ra_L\,\frac{\bfu^\bfn}{\bfn!}
\Bigr)\,. $$
Consequently, \eqref{eq:lrec} and ~\eqref{eq:Hindform} together with~\eqref{eq:cumutovol}
and Proposition~\ref{prop:EOconv} imply the claim.
\end{proof}

