Relations for quadratic Hodge integrals via stable maps

Abstract

Following Faber–Pandharipande, we use the virtual localization formula for the moduli space of stable maps to $\mathbb {P}^{1}$ to compute relations between Hodge integrals. We prove that certain generating series of these integrals are polynomials.

1 Introduction

Let $\overline {\mathcal {M}}_{g,n}$ be the moduli space of n-pointed genus g stable curves. It is a proper smooth Deligne Mumford (DM) stack of dimension $3g-3+n$ . We denote by $\pi :\overline {\mathcal {C}}_{g,n}\to \overline {\mathcal {M}}_{g,n}$ the universal curve and by $\sigma _i: \overline {\mathcal {M}}_{g,n}\to \overline {\mathcal {C}}_{g,n}$ the sections associated with the marking i for all $1\leq i\leq n$ . We denote by $\omega _{\overline {\mathcal {C}}_{g,n}/ \overline {\mathcal {M}}_{g,n}}$ the relative dualizing sheaf of $\pi $ . We will consider the following classes in $A^*(\overline {\mathcal {M}}_{g,n})$ :

• • For all $0\leq i\leq g$ , $\lambda _i$ stands for the ith Chern class of the Hodge bundle, i.e., the vector bundle $\mathbb {E}=\pi _*\omega _{\overline {\mathcal {C}}_{g,n}/ \overline {\mathcal {M}}_{g,n}}$ . For all $\alpha \in \mathbb {C}$ , we denote $\Lambda _g(\alpha )=\sum _{j=0}^g\alpha ^{g-j} \lambda _j$ , and $\Lambda ^\vee _g(\alpha )=(-1)^g\Lambda _g(-\alpha )$ . ¹

• • For all $1\leq i\leq n$ , we denote $\psi _i$ the Chern class of the cotangent line at the ith marking $\mathcal {L}_i=\sigma _i^*(\omega _{\overline {\mathcal {C}}_{g,n}/ \overline {\mathcal {M}}_{g,n}})$ .

A Hodge integral is an intersection number of the form:

$$ \begin{align*}\int_{\overline{\mathcal{M}}_{g,n}} \psi_1^{k_1}\ldots \psi_n^{k_n}\Lambda_g(t_1)\ldots \Lambda_g(t_m), \end{align*} $$

where $k_1,\ldots ,k_n$ are nonnegative integers and $t_1,\ldots ,t_m$ are complex numbers. If $m=1,2,$ or 3, then the above integral is called a linear, double, or triple Hodge integrals, respectively. Relations between linear Hodge integrals where proved in [FP00a] using the Gromov–Witten theory of $\mathbb {P}^1$ and the localization formula of [GP99]. This approach was also used in [FP00b] and [TZ03] to prove certain properties of triple Hodge integrals. Linear and triple Hodge integrals naturally appeared in the GW-theory of Calabi–Yau 3-folds, thus explaining a more abundant literature on the topic. However, double Hodge integrals have appeared recently in the quantization of Witten–Kontsevich generating series (see [Blo20]), in the theory of spin Hurwitz numbers (see [GKL21]), and in the GW theory of blow-ups of smooth surfaces (see [GKLS22]).

In the present note, we consider the following power series in $\mathbb {C}[\alpha ][\![t]\!]$ defined using double Hodge integrals:

$$ \begin{align*} P_a(\alpha ,t)=\sum_{g\geq0} t^{g}\left( \int_{\overline{\mathcal{M}}_{g,n+1}}\!\!\!\!\!\!\!\!\!\!\!\! \frac{\Lambda_{g}^{\vee}(1)\Lambda_{g}^{\vee}(\alpha )}{1-\psi_0}\prod_{i=1}^{n} {(2a_i+1)!!(-4\psi_i)^{a_i}}{}\right) \text{exp}\left(\frac{t}{24}\right), \end{align*} $$

where $a=(a_1,\ldots ,a_n)$ is a vector of nonnegative integers. If $n=1$ , we use the convention: $\int _{\overline {\mathcal {M}}_{0,2}} \psi _1^a \frac {\Lambda _{g}^{\vee }(1)\Lambda _{g}^{\vee }(\alpha ))}{1- \psi _2}=(-1)^a$ .

Theorem 1.1 $P_a(\alpha ,t)$ is a monic polynomial in $\mathbb {C}[\alpha ][t]$ of degree $|a|$ in t.

Here, we provide the first values of $P_a(-\alpha -1,t)$ . In the list below, we omit the variables $-\alpha -1$ and t in the notation:

$$ \begin{align*} &P_{()}= 1.\\[7pt] &P_{(1)}= t+12.\\[7pt] &P_{(2)}=t^2 - 10 \alpha t + 240. \\ &P_{(1,1)}=t^2 - 12 t. \\[7pt] &P_{(3)}=t^3 + (-77/3 \alpha - 28) t^2 + 280 t +6720. \\ &P_{(2,1)}= t^3 + (-10\alpha - 48)t^2 +(240\alpha + 240)t. \\ &P_{(1,1,1)}=t^3 - 72t^2+432t. \\[7pt] & P_{(4)}=t^4+ (-43\alpha - 72)t^3 + (126\alpha ^2 + 756\alpha + 840)t^2 + 10080t + 241920. \\ & P_{(3,1)}= t^4+ (-77/3 \alpha - 100) t^3+ (1232 \alpha + 1624) t^2. \\ & P_{(2,2)}=t^4+ (20\alpha + 100)t^3+(-100\alpha ^2 - 1360\alpha - 1680)t^2. \\ & P_{(2,1,1)}=t^4+ (-10\alpha - 132)t^3 + (840\alpha + 3120)t^2 +(-8640\alpha - 8640)t. \\ & P_{(1,1,1,1)}=t^4 - 168t^3 + 5616t^2 -20736t.\end{align*} $$

Considering these first values, we conjecture that $P_a$ is a polynomial of total degree $|a|$ in both variables t and $\alpha $ .

2 Preliminaries

We denote by $\overline {\mathcal {M}}_{g,n}(\mathbb {P}^1,1),$ the moduli space of stable maps of degree $1$ to $\mathbb {P}^1$ . It is a proper DM stack of virtual dimension $2g+n$ . Here, we can define in an analogous way the Hodge bundle $\mathbb {E}$ , the cotangent line bundles $\mathcal {L}_i$ and we denote again $\lambda _i$ and $\psi _i$ the respective Chern classes. We also have the forgetful and evaluation maps

$$ \begin{align*} \pi\colon \overline{\mathcal{M}}_{g,n+1}(\mathbb{P}^1,1)\to \overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,1),\ \text{and} \ \ \ ev_i\colon \overline{\mathcal{M}}_{g,n+1}(\mathbb{P}^1,1)\to \mathbb{P}^1. \end{align*} $$

Throughout this note, the enumeration of markings starts from $0$ . Furthermore, $\pi $ is the morphism that forgets the marking $p_0$ and $ev_i$ is the evaluation of a stable map to the ith marked point. The vector bundle $T:=R^1\pi _*(ev_0^*\mathcal {O}_{\mathbb {P}^1}(-1))$ is of rank g and we denote by y its top Chern class. We will denote:

$$ \begin{align*} \langle \prod_{i=0}^{n-1}\tau_{a_i}(\omega)|y\rangle_{g,1}^{\mathbb{P}^1}:= \int_{[\overline{\mathcal{M}}_{g,n}(\mathbb{P}^1,1)]^{vir}}\prod_{i=0}^{n-1}\psi_i^{a_i}ev_i^*(\omega)y, \end{align*} $$

where $\omega $ denotes the class of a point in $\mathbb {P}^1$ .

Theorem 2.1 (Localization Formula [GP99, FP00a])

Let $g\in \mathbb {Z}_{\geq 0}$ , and let $a\in \mathbb {Z}^n_{\geq 0}$ such that $|a|\leq g$ . Then, for all complex numbers ${\alpha }$ , and $t\in \mathbb {C}^*$ , we have

$$ \begin{align*} \langle \prod_{i=1}^n\tau_{a_i}(\omega)|y\rangle_{g,1}^{\mathbb{P}^1}= \sum_{g_1+g_2=g} &\int_{\overline{\mathcal{M}}_{g_1,n+1}}t^{n}\prod_{i=1}^n\psi_i^{a_i} \frac{\Lambda^{\vee}_{g_1}(t)\Lambda_{g_1}^{\vee}(\alpha t)}{t(t-\psi_0)}\ \\ &\times\int_{\overline{\mathcal{M}}_{g_2,1}} \frac{\Lambda_{g_2}^{\vee}(-t)\Lambda_{g_2}^{\vee}((\alpha +1)t)}{-t(-t-\psi_0)}. \end{align*} $$

Here, we use the convention $\int _{\overline {\mathcal {M}}_{0,1}}\psi _0^a=1$ .

Proposition 2.2 (Proposition 4.1 of [TZ03])

For all complex numbers $\alpha $ , we have

$$ \begin{align*} F(\alpha ,t)=1+\sum_{g>0}t^{2g}\int_{\overline{\mathcal{M}}_{g,1}}\frac{\Lambda_{g}^{\vee}(1)\Lambda_{g}^{\vee}(\alpha )}{1-\psi_0}=\text{exp} \left(-\frac{t^2}{24}\right). \end{align*} $$

Besides, we have the String and Dilaton equation for Hodge integrals.

Proposition 2.3 Let $g,n\in \mathbb {Z}_{\geq 0}$ such that $2g-2+n>0$ .

1. (i) [Dilaton equation for Hodge integrals] Let $(a_1,\dots ,a_{n})\in \mathbb {Z}_{\geq 0}^{n}$ and assume that there exist $i_0$ such that $a_{i_0}=1$ . Then

   $$ \begin{align*} \int_{\overline{\mathcal{M}}_{g,n+1}}\frac{\psi_{i_0}\prod_{i\neq i_0}\psi_i^{a_i}\prod_{j=1}^g\lambda_k^{b_k}}{1-\psi_0}= (2g-2+n) \int_{\overline{\mathcal{M}}_{g,n}}\frac{\prod_{i=1}^{n-1}\psi_i^{a_i}\prod_{j=1}^g\lambda_k^{b_k}}{1-\psi_0}. \end{align*} $$

2. (ii) [String equation for Hodge integrals] Let $(a_1,\dots ,a_{n})\in \mathbb {Z}_{\geq 0}^{n}$ and assume that there exist $i_0$ such that $a_{i_0}=0$ . Then we have

   $$ \begin{align*} \int_{\overline{\mathcal{M}}_{g,n+1}}\frac{\prod_{i=1}^n\psi_i^{a_i}\prod_{j=1}^g\lambda_k^{b_k}}{1-\psi_0}&= \int_{\overline{\mathcal{M}}_{g,n}}\frac{\prod_{i=1}^{n-1}\psi_i^{a_i}\prod_{j=1}^g\lambda_k^{b_k}}{1-\psi_0}\\&\quad + \sum_{j=1}^n\int_{\overline{\mathcal{M}}_{g,n}}\frac{\psi_j^{a_j-1}\prod_{i\neq j}\psi_i^{a_i}\prod_{k=1}^g\lambda_k^{b_k}}{1-\psi_0}. \end{align*} $$

3 The calculation

Note that the GW-invariant $\langle \prod _{i=1}^n\tau _{a_i} (\omega )|y\rangle _{g,1}^{\mathbb {P}^1}$ is 0 unless $|a|=g$ for dimensional reasons. Indeed, $\text {dim}_{\mathbb {C}}[\overline {\mathcal {M}}_{g,n} (\mathbb {P}^1,1)]^{\text {vir}}=2g+n$ and the cycle we are integrating is in codimension $g+|a|+n$ . Using the above localization formula, and Lemma 2.1 of [TZ03] the intersection number $\langle \prod _{i=1}^n\tau _{a_i}(\omega )|y\rangle _{g,1}^{\mathbb {P}^1}$ is expressed as

$$ \begin{align*} &\sum_{g_1+g_2=g}\int_{\overline{\mathcal{M}}_{g_1,n+1}}\!\!\!\!\!\!t^{n}\prod_{i=1}^n\psi_i^{a_i} \frac{\Lambda_{g_1}^{\vee}(t)\Lambda_{g_1}^{\vee}(\alpha t)}{t(t-\psi_0)}\ \cdot \int_{\overline{\mathcal{M}}_{g_2,1}} \frac{\Lambda^{\vee}_{g_2}(-t)\Lambda_{g_2}^{\vee}((\alpha +1)t)}{-t(-t-\psi_0)}\\ =& \sum_{g_1+g_2=g} t^{|a|-g_1}(-t)^{-g_2}\int_{\overline{\mathcal{M}}_{g_1,n+1}} \prod_{i=1}^{n}\psi^{a_i}\frac{\Lambda_{g_1}^{\vee}(1)\Lambda_{g_1}^{\vee}(\alpha )}{1-\psi_0} \times \int_{\overline{\mathcal{M}}_{g_2,1}}\frac{\Lambda_{g_2}^{\vee}(1)\Lambda_{g_2}^{\vee}(-(\alpha +1))}{1-\psi_0} \\ =& \ t^{|a|-g}\sum_{g_1+g_2=g}\int_{\overline{\mathcal{M}}_{g_1,n+1}} \prod_{i=1}^{n}\psi^{a_i}\frac{\Lambda_{g_1}^{\vee}(1)\Lambda_{g_1}^{\vee}(\alpha )}{1-\psi_0}\cdot \int_{\overline{\mathcal{M}}_{g_2,1}}\psi_0^{3g_2-2}. \end{align*} $$

In the last equation, we used Proposition 2.2 in order to replace $\int _{\overline {\mathcal {M}}_{g_2,1}}\frac {\Lambda _{g_2}^{\vee } (1)\Lambda _{g_2}^{\vee }(-(\alpha +1))}{1-\psi _0}$ with $(-1)^{g_2}\int _{\overline {\mathcal {M}}_{g_2,1}}\psi _0^{3g_2-2}$ .

We define

$$ \begin{align*} A_{g,a}(\alpha ) = \sum_{g_1+g_2=g}\int_{\overline{\mathcal{M}}_{g_1,n+1}} \prod_{i=1}^{n}\psi^{a_i}\frac{\Lambda_{g_1}^{\vee}(1)\Lambda_{g_1}^{\vee}(\alpha )}{1-\psi_0}\cdot \int_{\overline{\mathcal{M}}_{g_2,1}}\psi_0^{3g_2-2}. \end{align*} $$

Then, we have

$$ \begin{align*} A_{g,a}(\alpha )= \left\{\begin{matrix} 0, &|a|<g, \\ \langle \prod_{i=1}^n\tau_{a_i}(\omega)|y\rangle_{g,1}^{\mathbb{P}^1}, & |a|=g. \end{matrix} \right. \end{align*} $$

By the definition of $\Lambda _g^{\vee }(t),$ we see that $\Lambda _g^{\vee }(1)\Lambda _g^{\vee }(-(\alpha +1))$ is a polynomial in $\alpha $ of degree g, which actually determines the degree of $A_g(\alpha )$ .

We now present a proof for the main result.

Proof (of Theorem 1.1)

We begin by stating the well-known fact

$$ \begin{align*} 1 + \sum_{g>0}t^{g}\int_{\overline{\mathcal{M}}_{g,1}}\psi_0^{3g-2} =\exp\left({\frac{t}{24}}\right) \end{align*} $$

proven in Section 3.1 of [FP00a]. Now, we consider the product of $\exp \left ({\frac {t}{24}}\right )$ and

$$ \begin{align*} \sum_{g\geq0} t^{g}\left( \int_{\overline{\mathcal{M}}_{g,n+1}}\!\!\!\!\!\!\!\!\!\!\!\! \frac{\Lambda_{g}^{\vee}(1)\Lambda_{g}^{\vee}(\alpha )}{1-\psi_0}\prod_{i=1}^{n} {(2a_i+1)!!(-4\psi_i)^{a_i}}{}\right) \end{align*} $$

to obtain a new power series whose coefficients in degree g are given by

$$ \begin{align*} \sum_{g_1+g_2=g}\int_{\overline{\mathcal{M}}_{g_1,n+1}}\prod_{i=1}^{n}(2a_i+1)!!(-4)^{a_i} \prod_{i=1}^{n}\psi^{a_i}\frac{\Lambda_{g_1}^{\vee}(1)\Lambda_{g_1}^{\vee}(\alpha )}{1-\psi_0}\cdot \int_{\overline{\mathcal{M}}_{g_2,1}}\psi_0^{3g_2-2}. \end{align*} $$

This is exactly $A_{g,a}(\alpha )\cdot \prod _{i=1}^{n}(2a_i+1)!!(-4)^{a_i}$ . Hence, we can rewrite the power series $P_a(\alpha ,t)$ in the form

$$ \begin{align*} P_a(\alpha ,t)=\prod_{i=1}^{n}(2a_i+1)!!(-4)^{a_i}\sum_{g\geq0}t^{g}A_{g,a}(\alpha ). \end{align*} $$

As it is computed in the start of Section 3, we have that the numbers $A_{g,a}(\alpha )$ vanish when $g>|a|$ . Hence, we get that all coefficients of the power series $P_a(\alpha ,t)$ vanish when $g> |a|$ , i.e. $P_a(\alpha ,t)$ is a polynomial of degree $|a|$ . Furthermore, the top coefficient of $P_a(\alpha ,t)$ , i.e., the coefficient of $t^{|a|}$ is given by

$$ \begin{align*} \langle \prod_{i=1}^n (-4)^{a_i}(2a_i+1)!! \tau_{a_i}(\omega)|y\rangle_{|a|,1}^{\mathbb{P}^1}. \end{align*} $$

This value is computed in [KL11] and is actually equal to $1$ . In particular, the number $\prod _{i=1}^n (-4)^{a_i}(2a_i+1)!! $ is here to make the polynomial monic.

We now prove several other properties of the polynomials $P_a$ .

Proposition 3.1 The constant term $c_0$ of $P_a(\alpha ,t)$ is nonzero if and only if $n=1,$ where then $c_0=(-1)^a\prod _{i=1}^n (-4)^{a_i}(2a_i+1)!!$ or if $n>1$ and $\sum _{i=1}^na_i\leq n-2$ where then

$$ \begin{align*} c_0=\prod_{i=1}^n (-4)^{a_i}(2a_i+1)!!\frac{(n-2)!}{a_1!\dots (n-2-\sum{a_i})!}. \end{align*} $$

Proof We only compute the integrals appearing in the constant term of this polynomial since then we only have to multiply with $\prod _{i=1}^{n}(2a_i+1)!!(-4)^{a_i}$ . The integral in the constant term of $P_a(\alpha ,t)$ is given by $\int _{\overline {\mathcal {M}}_{0,n+1}}\frac {\prod _{i=1}^n\psi _i^{a_i}}{1-\psi _0}$ . When $n=1$ , using the convention $\int _{\overline {\mathcal {M}}_{0,2}}\frac {\psi _1^{a}}{1-\psi _0}=(-1)^a$ , we get that

$$ \begin{align*} c_0=(-1)^a\prod_{i=1}^n (-4)^{a_i}(2a_i+1)!!. \end{align*} $$

When $n>1$ , if $\sum _{i=1}^na_i>n-2$ , then $c_0$ is zero for dimensional reasons. Otherwise, we have

$$ \begin{align*} \hspace{-15pt}\int_{\overline{\mathcal{M}}_{0,n+1}}\frac{\prod_{i=1}^n\psi_i^{a_i}}{1-\psi_0}=\int_{\overline{\mathcal{M}}_{0,n+1}} \psi_0^{n-2-\sum a_i}\prod_{i=1}^n\psi_i^{a_i}= \frac{(n-2)!}{a_1!\dots (n-2-\sum{a_i})!}.\\[-42pt] \end{align*} $$

Proposition 3.2 Let $n\geq 3$ . Then we have the following rules:

1. (i) [String equation]

   $$ \begin{align*} P_{(a_1,\dots,a_{n-1},0)}(\alpha ,t)= P_{(a_1,\dots,a_{n-1})}(\alpha ,t)- \sum_{i=1}^n (8a_i+4) P_{(a_1,..,a_i-1,\dots,a_{n-1})}(\alpha ,t). \end{align*} $$

2. (ii) [Dilaton equation]

   $$ \begin{align*} P_{(a_1,\dots,a_{n-1},1)}(\alpha ,t)=(t-12n+24) P_{(a_1,\dots,a_{n-1})}(\alpha ,t) - 24 t P^{\prime}_{(a_1,\dots,a_{n-1})}(\alpha ,t)). \end{align*} $$

Proof We define the power series

$$ \begin{align*} \widetilde{P}_a(\alpha ,t)= \sum_{g\geq0} t^{g}\left(\int_{\overline{\mathcal{M}}_{g,n+1}}\prod_{i=1}^{n}\psi^{a_i} \frac{\Lambda_{g}^{\vee}(1)\Lambda_{g}^{\vee}(\alpha )}{1-\psi_0}\right). \end{align*} $$

Note that the following equation holds:

$$ \begin{align*} P_a(\alpha ,t)=\prod_{i=1}^{n}(2a_i+1)!!(-4)^{a_i}\widetilde{P}_a(\alpha ,t)\exp\left(\frac{t}{24}\right). \end{align*} $$

We can rewrite the coefficients of $\widetilde {P}_a(\alpha ,t)$ as

$$ \begin{align*} \sum_{k=0}^g\sum_{j=0}^g(-1)^{g+k}(a+1)^{g-j}\int_{\overline{\mathcal{M}}_{g,n+1}} \frac{\prod_{i=1}^n\psi_i^{a_i}\lambda_k\lambda_j}{1-\psi_0}. \end{align*} $$

1. (i) Applying the String equation for Hodge integrals, we obtain the following formula:

   $$ \begin{align*} \widetilde{P}_{(a_1,\dots,a_{n-1},0)}(\alpha ,t)=\widetilde{P}_{(a_1,\dots,a_{n-1})}(\alpha ,t)+ \sum_{i=1}^n\widetilde{P}_{(a_1,..,a_i-1,\dots,a_{n-1})}(\alpha ,t). \end{align*} $$

   Hence, multiplying with $\prod _{i=1}^{n-1}(2a_i+1)!!(-4)^{a_i} \text {exp}\left (\frac {t}{24}\right )$ , we obtain the desired result after a straightforward calculation.

2. (ii) Applying Dilaton equation for Hodge integrals, we obtain the following formula:

   $$ \begin{align*} \widetilde{P}_{(a_1,\dots,a_{n-1},1)}(\alpha ,t)&= 2\sum_{g\geq0}gt^g\int_{\overline{\mathcal{M}}_{g,n-1}}\prod_{i=1}^{n-1}\psi_i^{a_i}\frac{\Lambda^{\vee}_g(1)\Lambda^{\vee}_g(\alpha )}{1-\psi_0} \\ &+ (n-2)\widetilde{P}_{(a_1,\dots,a_{n-1})}(\alpha ,t). \end{align*} $$

   Note that the first term of the sum is equal to $2t\widetilde {P}^{\prime }_{(a_1,\dots ,a_{n-1})}(\alpha ,t)$ . Now, multiplying both sides of the equation above with

   $$ \begin{align*} \prod_{i=1}^{n-1}(2a_i+1)(-4)^{a_i}\text{exp}\left(\frac{t}{24}\right), \end{align*} $$

   we have

   $$ \begin{align*} \frac{-1}{12}P_{(a_1,\dots,a_{n-1},1)}(\alpha ,t)&=(n-2)P_{(a_1,\dots,a_{n-1})}(\alpha ,t)\\ & + 2t \left(\prod_{i=1}^{n-1} (-4)^{a_i}(2a_i+1)!!\right) \widetilde{P}^{\prime}_{(a_1,\dots,a_{n-1})}(\alpha ,t)\mathrm{e}^{{t}/{24}}\\ \!\! &= (n-2)P_{(a_1,\dots,a_{n-1})}(\alpha ,t)\\ & + 2t( P^{\prime}_{(a_1,\dots,a_{n-1})}(\alpha ,t)-\frac{1}{24} P_{(a_1,\dots,a_{n-1})}(\alpha ,t)). \end{align*} $$

   Finally, clearing the denominators, we obtain the desired result.

We recall Mumford’s relation $\Lambda ^\vee _g(1)\cdot \Lambda ^\vee _g(-1)=1$ (see [Mum83]). In particular, $P_a(-1,t)$ is defined by integrals of $\psi $ -classes.

Corollary 3.3 For any vector $a\in \mathbb {Z}^n_{\geq 0}$ , the power series

$$ \begin{align*} P_a(-1,t)=\prod_{i=1}^{n}(2a_i+1)!!(-4)^{a_i}\text{exp}\left(\frac{t}{24}\right) \cdot\sum_{g\geq0}(-t)^g\int_{\overline{\mathcal{M}}_{g,n+1}} \frac{\prod_{i=1}^n\psi_i^{a_i}}{1-\psi_0} \end{align*} $$

is a polynomial of degree $|a|$ .

In this case, the polynomiality as well as a closed expression were proved in [LX11].