Series and integral representations of the Taylor coefficients of the Weierstrass sigma-function

TL;DR

The paper introduces new series and integral formulas for the Taylor coefficients of the Weierstrass sigma-function associated with any lattice in the complex plane, enhancing analytical tools for elliptic functions.

Contribution

It provides novel Hermite-Gauss series and integral representations for the sigma-function's Taylor coefficients, applicable to arbitrary complex lattices.

Findings

Derived Hermite-Gauss series representation for coefficients

Established Hermite-Gauss integral formulas over complex plane

Applicable to any lattice in the complex plane

Abstract

We provide two kinds of representations for the Taylor coefficients of the Weierstrass $\sigma$-function $\sigma(\cdot;\Gamma)$ associated to an arbitrary lattice $\Gamma$ in the complex plane $\mathbb{C}=\mathbb{R}^2$ - the first one in terms of the so-called Hermite-Gauss series over $\Gamma$ and the second one in terms of Hermite-Gauss integrals over $\mathbb{C}$.

Equations

$$\sigma(z;\Gamma)=z\prod_{\gamma\in\Gamma\setminus\{0\}}\Big{(}1-\frac{z}{\gamma}\Big{)}e^{\frac{z}{\gamma}+\frac{1}{2}\left(\frac{z}{\gamma}\right)^{2}}.$$

$$\sigma (z;\Gamma )=\sum _{r=0}^{\infty }{\mathcal{W}}_r(\Gamma )\frac{z^{2r+1}}{(2r+1)!}.$$

$$g_2=60\sum _{\gamma \in \Gamma \setminus \{0\}}\frac{1}{{\gamma }^4}\text{mbox}and{g}_3=140\sum _{\gamma \in \Gamma \setminus \{0\}}\frac{1}{{\gamma }^6},$$

$${\mathscr{W}}_{r}(\Gamma):=\sum_{\begin{array}[]{c}2m+3n=r\\ m,n\geq 0\end{array}}a_{m,n}\left(\frac{g_{2}}{2}\right)^{m}(2g_{3})^{n},$$

$$a_{m,n}=3(m+1)a_{m+1,n-1}$$

$$-\frac{1}{3}(2m+3n-1)(4m+6n-1)a_{m-1,n}.$$

$$\sigma (z;\Gamma )$$

$$-\frac{g_2{g}_3}{2^7\cdot 3^2\cdot 5^2\cdot 7\cdot 11}{z}^{11}+\frac{(23g_2^3-576g_3^2)}{2^{10}\cdot 3^4\cdot 5^2\cdot 7\cdot 11\cdot 13}{z}^{13}+\cdots .$$

$$\nu =\nu (\Gamma )=\frac{\pi }{S_{\Gamma }}$$

$$\mu =\mu (\Gamma )=\frac{i}{S_{\Gamma }}\left(\zeta \left(\frac{{\omega }_1}{2};\Gamma \right){\overline{\omega }}_2-\zeta \left(\frac{{\omega }_2}{2};\Gamma \right){\overline{\omega }}_1\right).$$

$$\chi_{{}_{W}}(\gamma)=\left\{\begin{array}[]{ll}+1&\qquad\mbox{if}\quad\gamma/2\in\Gamma\\ -1&\qquad\mbox{if}\quad\gamma/2\not\in\Gamma\end{array}\right..$$

$${\mathcal{H}}_r(\Gamma )=\nu \frac{(2r+1)!}{2^r\cdot r!}\sum _{\gamma \in \Gamma }{\mu }^r\mathbf{F}\left(-r;\frac{3}{2};-\frac{{\nu }^2{\overline{\gamma }}^2}{2\mu }\right){\chi }_{{}_W}(\gamma )|\gamma {|}^2{e}^{-\frac{\nu }{2}|\gamma {|}^2}$$

$${\mathcal{H}}_r(\Gamma )=\sum _{\gamma \in \Gamma }{\left(\sqrt{-\frac{\mu }{2}}\right)}^{2r+1}{H}_{2r+1}\left(\frac{\nu \overline{\gamma }}{\sqrt{-2\mu }}\right){\chi }_{{}_W}(\gamma )\gamma e^{-\frac{\nu }{2}|\gamma {|}^2}.$$

$$\sigma (z;\Gamma )=\frac{1}{{\mathcal{H}}_o(\Gamma )}{e}^{\frac{\mu }{2}{z}^2}\sum _{\gamma \in \Gamma }{\chi }_{{}_W}(\gamma )\gamma e^{-\frac{\nu }{2}|\gamma {|}^2+\nu z\bar{\gamma }}.$$

$$\sigma (z;\Gamma )=\left(\frac{\nu }{\pi }\right)e^{\frac{\mu }{2}{z}^2}{\int }_{\mathbb{C}}{e}^{\nu z\overline{w}-\frac{\mu }{2}{w}^2-\nu |w{|}^2}\sigma (w;\Gamma )dm(w),$$

$${\mathcal{W}}_r(\Gamma )=\frac{{\mathcal{H}}_r(\Gamma )}{{\mathcal{H}}_o(\Gamma )}.$$

$${\mathcal{W}}_r(\Gamma )$$

$$\times {\int }_{\mathbb{C}}\overline{w}{\mu }^r\mathbf{F}\left(-r;\frac{3}{2};-\frac{{\nu }^2{\overline{w}}^2}{2\mu }\right)e^{-\frac{\mu }{2}{w}^2-\nu |w{|}^2}\sigma (w;\Gamma )dm(w).$$

$${\mathcal{H}}_o(\Gamma )=\nu \sum _{\gamma \in \Gamma }{\chi }_{{}_W}(\gamma )|\gamma {|}^2{e}^{-\frac{\nu }{2}|\gamma {|}^2}$$

$$f(z+\gamma ):=\chi (\gamma )e^{\frac{\nu }{2}|\gamma {|}^2+\nu z\overline{\gamma }}f(z).$$

$${\mathcal{O}}_{\Gamma ,\chi }^{\nu }(\mathbb{C})=\left\{f\text{mbox}entire;f(z+\gamma )=\chi (\gamma )e^{\frac{\nu }{2}|\gamma {|}^2+\nu z\overline{\gamma }}f(z),z\in \mathbb{C},\gamma \in \Gamma \right\},$$

$$\chi (\gamma +{\gamma }^{\prime })=\chi (\gamma )\chi ({\gamma }^{\prime })e^{\frac{\nu }{2}(\gamma \overline{{\gamma }^{\prime }}-\overline{\gamma }{\gamma }^{\prime })},\gamma ,{\gamma }^{\prime }\in \Gamma .$$

$$\dim {\mathcal{O}}_{\Gamma ,\chi }^{\nu }(\mathbb{C})=\left(\frac{\nu }{\pi }\right)S_{\Gamma },$$

$${⟨⟨f,g⟩⟩}_{\Gamma }={\int }_{\Lambda (\Gamma )}f(z)\overline{g(z)}{e}^{-\nu |z{|}^2}dm(z)$$

$${K}^{\nu}_{\Gamma,\chi}(z,w):=\left(\frac{\nu}{\pi}\right)e^{\nu z\overline{w}}\sum_{\gamma\in\Gamma}\chi(\gamma)e^{-\frac{\nu}{2}|\gamma|^{2}+\nu\big{(}z\bar{\gamma}-\overline{w}\gamma\big{)}}.$$

$${\int }_{\Lambda (\Gamma )}{K}_{\Gamma ,\chi }^{\nu }(z,z)e^{-\nu |z{|}^2}dm(z)=\left(\frac{\nu }{\pi }\right)S_{\Gamma }\ne 0.$$

$$K_{\Gamma ,\chi }^{\nu }(z+\gamma ,w+{\gamma }^{\prime })=\chi (\gamma )e^{\frac{\nu }{2}|\gamma {|}^2+\nu z\overline{\gamma }}{K}_{\Gamma ,\chi }^{\nu }(z,w)\overline{\chi ({\gamma }^{\prime })}{e}^{\frac{\nu }{2}|{\gamma }^{\prime }{|}^2+\nu \overline{w}{\gamma }^{\prime }}$$

$$f(z)=\left(\frac{\nu }{\pi }\right){\int }_{\mathbb{C}}{e}^{\nu z\overline{w}}f(w)e^{-\nu |w{|}^2}dm(w)$$

$$f(z)={\int }_{\Lambda (\Gamma )}{K}_{\Gamma ,\chi }^{\nu }(z,w)f(w)e^{-\nu |w{|}^2}dm(w).$$

$${\int }_{\Lambda (\Gamma )}{K}_{\Gamma ,\chi }^{\nu }(z,z)e^{-\nu |z{|}^2}dm(z)$$

$${\int }_{\Lambda (\Gamma )}{e}^{\nu (z\bar{\gamma }-\bar{z}\gamma )}dm(z)=0$$

$$K_{\Gamma ,\chi }^{\nu }(z+\gamma ,w)=\chi (\gamma )e^{\frac{\nu }{2}|\gamma {|}^2+\nu z\bar{\gamma }}{K}_{\Gamma ,\chi }^{\nu }(z,w).$$

Full text

1. Introduction and statement of main results

We let $\Gamma\subset\mathbb{C}$ be a given two-dimensional lattice and denote by $\sigma(z;\Gamma)$ the associated Weierstrass $\sigma$-function defined through the convergent infinite product

[TABLE]

It is an odd entire function on $\mathbb{C}$ and therefore its Taylor series expansion at $z=0$ can be written as

[TABLE]

A representation of the Taylor coefficient ${\mathscr{W}}_{r}(\Gamma)$, in terms of the Eisenstein series

[TABLE]

has been obtained by K. Weierstrass [12] in 1882, to wit

[TABLE]

where the coefficients $a_{m,n}$ are such that $a_{0,0}=1$ and $a_{m,n}=0$ whenever $m<0$ or $n<0$. Otherwise, they are connected by the recursion formula

[TABLE]

The first few values of the coefficients ${\mathscr{W}}_{r}(\Gamma)$ in (1.4); up to $r\leq 6$, are known to be given by [8, page 391]

[TABLE]

In the present paper, we shall give new representations of the Taylor coefficients ${\mathscr{W}}_{r}(\Gamma)$ either as series over $\Gamma$ of Hermite-Gauss type, or as integrals over $\mathbb{C}$ of the $\sigma$-function. Before doing so, we fix the notations we shall use repeatedly in the sequel: Let $\left\{\omega_{1},\omega_{2}\right\}$ be an oriented arbitrary $\mathbb{R}$-basis of $\mathbb{C}=\mathbb{R}^{2}$ and set $\Gamma=\mathbb{Z}\omega_{1}+\mathbb{Z}\omega_{2}$. The area of a fundamental cell $\Lambda(\Gamma)$ of $\Gamma$ is given by $S=S_{\Gamma}=\Im(\overline{\omega_{1}}\omega_{2})$, where $\Im(z)$ denotes the imaginary part of $z\in\mathbb{C}$. We define $\nu\in\mathbb{R}^{>}:=]0,+\infty[$ and $\mu\in\mathbb{C}$ by

[TABLE]

and

[TABLE]

Here $\zeta(z;\Gamma)={\sigma^{{}^{\prime}}(z;\Gamma)}/{\sigma(z;\Gamma)}$ is the Weierstrass zeta-function (not to be confused with the Riemann zeta-function). Also, let $\chi_{{}_{W}}$ be the “Weierstrass semi-character" defined on $\Gamma$ by

[TABLE]

Associated to the data of $\nu$, $\mu$ and $\chi_{{}_{W}}$, we consider the series

[TABLE]

for every nonnegative integer $r$. In (1.9), ${\mathbf{F}}(a;c;x)$ denotes the usual confluent hypergeometric function and $|\gamma|^{2}$ the square of the Euclidean length of the vector $\gamma\in\mathbb{C}=\mathbb{R}^{2}$. Note that (1.9) can be expressed in terms of the Hermite polynomial $H_{2r+1}$ of degree $2r+1$ as

[TABLE]

Definition 1.1**.**

Since the quantity ${\mathscr{H}}_{r}(\Gamma)$, as given by (1.10), involves the Hermite polynomial $H_{2r+1}$ as well as the Gaussian function $e^{-\frac{\nu}{2}|\gamma|^{2}}$, we may call it Hermite-Gauss series over $\Gamma$ of order $r$.

With the above notations, the main results of the present paper can be stated as follows:

Main Theorem 1.2**.**

Let $\Gamma$ be a lattice in $\mathbb{C}$, and keep $\nu=\nu(\Gamma)$, $\mu=\mu(\Gamma)$, $\chi_{{}_{W}}=\chi_{{}_{W}}(\Gamma)$ and ${\mathscr{H}}_{r}(\Gamma)$ as above. Then, we have

• i)

An expansion series over $\Gamma$ of the Weierstrass $\sigma$-function $\sigma(z;\Gamma)$:

[TABLE]

• ii)

A reproducing integral formula over $\mathbb{C}$ for $\sigma(z;\Gamma)$:

[TABLE]

where $dm$ denotes the usual Lebesgue measure on $\mathbb{C}$.

• iii)

A representation of the Taylor coefficient ${\mathscr{W}}_{r}(\Gamma)$ in terms of Hermite-Gauss series over $\Gamma$:

[TABLE]

• iv)

A representation of ${\mathscr{W}}_{r}(\Gamma)$ in terms of the Hermite-Gauss integral over $\mathbb{C}$:

[TABLE]

Remark 1.3**.**

It is likely impossible to proof directly that the quantity

[TABLE]

appearing in the denominator of (1.11) and (1.13), is not zero. However, we reassure the reader that ${\mathscr{H}}_{o}(\Gamma)\neq 0$ and a proof will be given in Section 4.

Remark 1.4**.**

The approach we will be adopting, to obtain representations (1.11) and (1.13), is based neither on the partial differential equations satisfied by $\sigma(z;\Gamma)$ and involving $g_{2}$ and $g_{3}$, as elaborated by Weierstrass in [12], nor on the bilinear operators as done in [3]. In fact, we will derive them from a concrete description of the functional space ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$ of $(\Gamma,\chi)$-theta functions of magnitude $\nu$ on $\mathbb{C}$, i.e., the space of all entire functions $f$ on $\mathbb{C}$ satisfying the functional equation

[TABLE]

As applications of Theorem 1.2, one can obtain highly nontrivial identities of arithmetic type and relate some modular forms to Hermite-Gauss series. Moreover, full series of identities including those obtained by Perelomov [10] can be deduced. Furthermore, our main results can be applied to show that, for every fixed integer $n\geq 2$, the Eisenstein series $G_{2n}(\Gamma)=\sum^{\prime}1/\gamma^{2n}$ can be expressed as a rational fraction in the derivatives $T_{j}=\widetilde{\theta_{{}_{W}}}^{(2j-1)}(0;\Gamma)$; $j=1,\cdots,n$, where $\widetilde{\theta_{{}_{W}}}(z;\Gamma)$ is the Weierstrass-theta series as defined by (3.6). The precise statements of the above mentioned applications as well as their details will be given in a separate paper.

The paper is outlined as follows. In Section 2, we review some basic properties of the Hilbert space ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$ of $(\Gamma;\chi)$-theta functions. In Section 3, we introduce the modified Weierstrass $\sigma$-function and the Weierstrass $(\Gamma;\chi)$-theta series, which have the important property of being generators of ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$. Section 4 deals with the proof of the main results summarized as Theorem 1.2.

2. Background on $(\Gamma,\chi)$-theta entire functions

In this section, we review some basic properties of the space ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$ of $(\Gamma,\chi)$-theta entire functions of magnitude $\nu>0$, i.e., the functional space

[TABLE]

where $\Gamma$ is a lattice in $\mathbb{C}$ and $\chi$ a given map on $\Gamma$ with $|\chi(\gamma)|=1$ and satisfying the (RDQ)-condition

[TABLE]

Remark 2.1**.**

By $(RDQ)$ we refer to the “Riemann-Dirac Quantization" condition for the pair $(H,E)$, with $H(z,w)=(\nu/\pi)\left<z,w\right>$ and $E(z,w)=\Im H(z,w)$.

The $({RDQ})$ condition is in fact a necessary and sufficient condition ensuring that ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$ is a nonzero vector space [5]. Indeed, $(RDQ)$ is equivalent to the condition that the complex valued function $J_{\nu,\chi}$ defined on $\Gamma\times\mathbb{C}^{n}$ by $J_{\nu,\chi}(\gamma,z):=\chi(\gamma)e^{i\nu E(z,\gamma)}$ is an automorphy factor satisfying the cocycle identity, $J_{\nu,\chi}(\gamma_{1}+\gamma_{2},z)=J_{\nu,\chi}(\gamma_{1},z+\gamma_{2})J_{\nu,\chi}(\gamma_{2},z)$. Therefore, $\gamma(z;v):=(z+\gamma;\chi(\gamma)e^{\frac{\nu}{2}|\gamma|^{2}+\nu z\overline{\gamma}}.v)$ defines an action of $\Gamma$ on $\mathbb{C}^{n}\times\mathbb{C}$. The associated quotient space $L:=(\mathbb{C}^{n}\times\mathbb{C})/\Gamma$ is a holomorphic line bundle over the torus $\mathbb{C}^{n}/\Gamma$ with fiber $\mathbb{C}=\tau^{-1}([z])$, where the projection map $\tau:(\mathbb{C}^{n}\times\mathbb{C})/\Gamma\longrightarrow\mathbb{C}^{n}/\Gamma$ is the natural one induced from the canonical projection $\pi:\mathbb{C}^{n}\longrightarrow\mathbb{C}^{n}/\Gamma$. Whence, the space ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$ can be viewed as the space of holomorphic sections of the holomorphic line bundle $L=(\mathbb{C}\times\mathbb{C})/\Gamma$. Its dimension is then given by the Pfaffian $\sqrt{\det E}$ of the associated skew-symmetric form $E$ (see [9, 7, 4, 2, 11]). In our case, $E(z,w):=(\nu/\pi)\Im\left<z,w\right>$. Therefore, we have the following

Proposition 2.2**.**

Let $\Gamma$ be a given lattice in $\mathbb{C}$, $\chi$ a map on $\Gamma$ with $|\chi(\gamma)|=1$ and $\nu>0$ such that the (RDQ)-condition holds. Then, the functional space ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$ is a finite dimensional space whose the dimension is given explicitly by

[TABLE]

where $S_{\Gamma}$ is the area of a fundamental cell of $\Gamma$.

Remark 2.3**.**

Note that for given $f,g\in{\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$, the function $z\mapsto f(z)\overline{g(z)}e^{-\nu|z|^{2}}$ is $\Gamma$-periodic. Thus, one can equip ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$ with the hermitian scalar product

[TABLE]

which turns out to be a positive definite hermitian inner product on ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$. Thus the finite dimensional Hilbert space ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$ is a reproducing kernel Hilbert space. The next result gives an explicit expression for its reproducing kernel.

Consider the kernel function ${K}^{\nu}_{\Gamma,\chi}(z,w)$ on $\mathbb{C}\times\mathbb{C}$ given by the series

[TABLE]

Then ${K}^{\nu}_{\Gamma,\chi}(z,w)$ is well defined as a holomorphic function in $z$ and antiholomorphic in $w$. Moreover, the kernel function ${K}^{\nu}_{\Gamma,\chi}(z,w)$ possesses the properties summarized below.

Theorem 2.4**.**

Let ${K}^{\nu}_{\Gamma,\chi}(z,w)$ be as in (2.3).

• i)

For arbitrary fundamental cell $\Lambda(\Gamma)$ of the lattice $\Gamma$, we have

[TABLE]

• ii)

The function ${K}^{\nu}_{\Gamma,\chi}(z,w)$ satisfies the $\Gamma$-bi-invariance property

[TABLE]

for every $z,w\in\mathbb{C}$ and $\gamma,\gamma^{\prime}\in\Gamma$.

• iii)

Every $f\in{\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$ can be reproduced either as

[TABLE]

or as

[TABLE]

A sketched proof of the above theorem is presented below. However, a detailed one can be found in [5, Theorem 3.7].

Sketched proof of Theorem 2.4. i) follows by direct computation of

[TABLE]

using (2.3) and taking into account the $(RDQ)$ condition which implies

[TABLE]

for every nonzero $\gamma\in\Gamma.$

To prove ii), we begin by writing an expression for ${K}^{\nu}_{\Gamma,\chi}(z+\gamma,w)$ using (2.3). Making a standard change of variable in the obtained series together with the $(RDQ)$ condition, we get

[TABLE]

This gives rise to ii) since ${K}^{\nu}_{\Gamma,\chi}(z,w)=\overline{{K}^{\nu}_{\Gamma,\chi}(w,z)}$.

For a proof of iii), we note first that for every given $f\in{\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$, there exists a certain constant $C>0$ such that $|f(z)|\leq Ce^{\frac{\nu}{2}|z|^{2}}$. Then by considering

[TABLE]

for $0<\varepsilon<1$, it follows that the holomorphic function $f_{\varepsilon}(z)$ satisfies the growth condition $|f_{\varepsilon}(z)|\leq Ce^{\frac{\nu}{2}\varepsilon^{2}|z|^{2}}$. Hence, it belongs to the Bargmann-Fock space $\mathcal{B}^{2,\nu}(\mathbb{C})=\mathcal{O}(\mathbb{C})\cap L^{2}(\mathbb{C};e^{-\nu|z|^{2}}dm(z)),$ whose the reproducing kernel is $(\nu/\pi)e^{\nu z\overline{w}}$ [1]. Therefore, we have

[TABLE]

Thus formula (2.6) follows by letting $\varepsilon\longrightarrow 1^{-}$ and applying the Lebesgue’s dominated convergence theorem. While (2.7) can be obtained from (2.6) by writing $\mathbb{C}$ as a disjoint union of $\gamma+\Lambda(\Gamma)$; $\gamma\in\Gamma$, and then using the fact that $f$ belongs to ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$. $\square$

Corollary 2.5**.**

The function ${K}^{\nu}_{\Gamma,\chi}(z,w)$ given by (2.3) is the reproducing kernel of the Hilbert space ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$.

Corollary 2.6**.**

There exists at least one point $w_{0}\in\mathbb{C}$ such that the map $z\mapsto{K}^{\nu}_{\Gamma,\chi}(z,w_{0})$ is not identically zero.

Proof. In view of the bi-invariance property (2.5), it is clear that for every fixed $w\in\mathbb{C}$ the entire function $z\mapsto{K}^{\nu}_{\Gamma,\chi}(z,w)$ belongs to ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$. The fact that $z\mapsto{K}^{\nu}_{\Gamma,\chi}(z,w_{0})$ is not identically zero for some $w_{0}\in\mathbb{C}$ follows from i) of Theorem 2.4. $\square$

Remark 2.7**.**

According to the proof provided above, the existence of $w_{0}\in\mathbb{C}$, for which $z\mapsto{K}^{\nu}_{\Gamma,\chi}(z,w_{0})$ is non identically zero, is eventually not effective. Moreover, Remark 3.4 below shows that $\omega_{0}$ belongs to $\mathbb{C}\setminus\Gamma$.

3. Generators of ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$ associated to the Weierstrass semi-character

From now on, we restrict ourselves to the particular case of the triplet $(\Gamma,\nu,\chi)$, where $\Gamma$ is a fixed lattice in $\mathbb{C}$, $\nu=\nu(\Gamma)=\pi/S_{\Gamma}$ and $\chi=\chi_{{}_{W}}(\gamma)$ is the Weierstrass semi-character as in (1.8),

[TABLE]

The following result is immediate (whose the proof is left to the reader):

Lemma 3.1**.**

Let $\nu=\nu(\Gamma)=\pi/S_{\Gamma}$ as above. Then $\chi=\chi_{{}_{W}}$ in (3.1) satisfies the $(RDQ)$ condition

[TABLE]

From Lemma 3.1 above and Theorem 2.4, we deduce that ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$ is a one dimensional vector space. Two kinds of generators for ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$ are constructed below - the first one can be considered as a modified Weierstrass $\sigma$-function, while the second one is a modified theta function. Its construction is based on the data of the reproducing kernel ${K}^{\nu}_{\Gamma,\chi}(z,w)$ of ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$.

3.1. The modified Weierstrass $\sigma$-function.

The main result of this section is the following

Theorem 3.2**.**

There exist a unique real number $\nu>0$ and a unique complex number $\mu$ such that the entire function

[TABLE]

is a generator of ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C}).$ More precisely, $\nu$ and $\mu$ are explicitly given by

[TABLE]

Definition 3.3**.**

The function $\widetilde{\sigma}(z;\Gamma)=\widetilde{\sigma}_{\mu}(z;\Gamma)$ defined by (3.3) and associated to the special value of $\mu$ given through (3.4) will be called the modified Weierstrass $\sigma$-function.

Proof of Theorem 3.2. We have to look for necessary and sufficient conditions on the pair $(\nu,\mu)$ so that the function $\widetilde{\sigma}_{\mu}(z;\Gamma)$, as given by (3.3), satisfies the functional equation

[TABLE]

For this, we begin by recalling the pseudo-periodicity [7, 6, 11], satisfied by the standard Weierstrass $\sigma$-function,

[TABLE]

where $\eta(\gamma)$ is defined by $\eta(m\omega_{1}+n\omega_{2})=m\eta_{1}+n\eta_{2}$, where $\{\omega_{1},\omega_{2}\}$ is a given oriented $\mathbb{R}$-basis of the lattice $\Gamma=\mathbb{Z}\omega_{1}+\mathbb{Z}\omega_{2}$ and $\eta_{j}$; $j=1,2$, are related to the Weierstrass zeta-function by $\eta_{j}=2\zeta(\omega_{j}/2)$. Therefore, $\widetilde{\sigma}_{\mu}(z;\Gamma)$ satisfies (3.5) if and only if $e^{(\mu\gamma+\nu\overline{\gamma}-\eta)(z+\gamma/2)}=1$ for every $z\in\mathbb{C}$ and every $\gamma\in\Gamma$. It follows then that the numbers $\nu$ and $\mu$ verify the linear system

[TABLE]

whose the determinant $\overline{\omega}_{1}\omega_{2}-\omega_{1}\overline{\omega}_{2}=2iS_{\Gamma}$ is not [math]. Whence,

[TABLE]

and therefore, the numbers $\nu$ and $\mu$ are explicitly given by

[TABLE]

The expression of $\nu$ reduces further to $\nu=\pi/{S_{\Gamma}}$, making use of the Legendre’s relation, $\eta_{1}\omega_{2}-\eta_{2}\omega_{1}=2i\pi$ ([6, page 102]). For such values of $\nu$ and $\mu$, the nonzero function $\widetilde{\sigma}_{\mu}(z;\Gamma)$ belongs to the one dimensional space ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$ and hence is a generator of ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$. $\square$

Remark 3.4**.**

Since ${K}^{\nu}_{\Gamma,\chi}(z,w)$, in (2.3), is the reproducing kernel of the one dimensional Hilbert space $\mathcal{O}^{\nu}_{\Gamma,\chi}(\mathbb{C})$ (Corollary 2.5), it follows that

[TABLE]

with $\left\|\widetilde{\sigma}_{\mu}(\cdot,\Gamma)\right\|^{-2}=-\nu^{2}{\mathscr{H}}_{o}(\Gamma)/\pi$, so that ${\mathscr{H}}_{o}(\Gamma)<0$. Whence, the set of zeros of $(z,w)\mapsto{K}^{\nu}_{\Gamma,\chi}(z,w)$ is exactly $(\Gamma\times\mathbb{C})\cup(\mathbb{C}\times\Gamma)$.

3.2. The Weierstrass-theta series.

Below, we consider another generator of ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$ given as series over $\Gamma$. Namely, we assert the following

Theorem 3.5**.**

Let $\nu=\pi/S_{\Gamma}$ and $\chi=\chi_{{}_{W}}$ as above. Then, the entire function $\widetilde{\theta_{{}_{W}}}(z;\Gamma)$ defined by

[TABLE]

is a generator of the space ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$.

Definition 3.6**.**

The odd entire function $\widetilde{\theta_{{}_{W}}}(z;\Gamma)$ given by (3.6) will be called the Weierstrass $(\Gamma,\chi)$-theta function.

To prove Theorem 3.5, we make use of the following lemma.

Lemma 3.7**.**

Let $\nu=\pi/S$ and $\chi=\chi_{{}_{W}}$ as above. Let $f$ be a holomorphic function on $\mathbb{C}$ such that $|f(z)|\leq Ce^{\alpha|z|^{\beta}}$ for some constant $C\geq 0$ and given real numbers $\alpha\geq 0$ and $\beta<2$. Define

[TABLE]

to be the periodization à la Poincaré of the function $f$. Then,

• i)

${\mathscr{P}}^{\nu}_{\Gamma,\chi}(f)$* belongs to ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$.*

• ii)

If in addition $f$ is an even function, then ${\mathscr{P}}^{\nu}_{\Gamma,\chi}(f)$ is also even and consequently is identically zero.

Proof of Theorem 3.5. By specifying $f(z)=1$ and appealing to ii) of Lemma 3.7, we easily get the following identity,

[TABLE]

for every $z\in\mathbb{C}$. Hence, it follows that

[TABLE]

and therefore $\widetilde{\theta_{{}_{W}}}(z;\Gamma)$ belongs to ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$, according to i) of Lemma 3.7. The hard part is to prove that $\widetilde{\theta_{{}_{W}}}(z;\Gamma)$ is not identically zero on $\mathbb{C}$. To handle this, we recall from Corollary 2.6 that the holomorphic function

[TABLE]

belonging to ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$, is not identically zero on $\mathbb{C}$ for some $w_{0}$. This means that $z\mapsto K^{\nu}_{\Gamma,\chi}(z,w_{0})$ is a generator of the one dimensional space ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$. Since the modified Weierstrass function $\widetilde{\sigma}(z;\Gamma)$ is also a generator of ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$, according to Theorem 3.2, there exists a constant $C_{w_{0}}\neq 0$ such that $K^{\nu}_{\Gamma,\chi}(z,w_{0})=C_{w_{0}}\widetilde{\sigma}(z;\Gamma)$. More explicitly, we have

[TABLE]

Differentiating both sides of (3.10) at $z=0$ and using the fact $\sigma^{\prime}(0;\Gamma)=1$ yield

[TABLE]

Now, taking the conjugate and changing $\gamma$ to $-\gamma$, keeping in mind that the Weierstrass pseudo-character $\chi$ is real and even, we obtain

[TABLE]

Finally, by means of (3.9), we deduce that $\widetilde{\theta_{{}_{W}}}(w_{0};\Gamma)=-\frac{\pi}{\nu^{2}}\overline{C_{w_{0}}}\neq 0$. This completes the proof of Theorem 3.5. $\square$

Sketched proof of Lemma 3.7. The growth condition $|f(z)|\leq Ce^{\alpha|z|^{\beta}}$, $\alpha\geq 0$, $\beta<2$, satisfied by the holomorphic function $f$, ensures that the Poincaré series (3.7) converges absolutely and uniformly on compact subsets of $\mathbb{C}$ and then is holomorphic. To conclude i), note that for every given $\gamma_{0}\in\Gamma$, we have

[TABLE]

The change of summation index $\gamma=\gamma_{0}+\gamma^{\prime}$ together with the $(RDQ)$ condition,

[TABLE]

yields

[TABLE]

Now, since $e^{\nu(\gamma_{0}\overline{\gamma^{\prime}}-\overline{\gamma_{0}}\gamma^{\prime})}=1$, which is an immediate consequence of the $(RDQ)$ condition, one deduces

[TABLE]

To get ii), we let $f$ be an even function satisfying the hypothesis of Lemma 3.7. Since $\chi$ is even, it is easy to show that ${\mathscr{P}}^{\nu}_{\Gamma,\chi}(f)$ is also an even function belonging to the one dimensional space ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$ which is generated by the odd function $\widetilde{\sigma}(z;\Gamma)$. So necessarily ${\mathscr{P}}^{\nu}_{\Gamma,\chi}(f)\equiv 0$ on $\mathbb{C}$. $\square$

We conclude this section with the following remark.

Remark 3.8**.**

Assertion ii) of Lemma 3.7 can be used to derive a full series of identities when specifying the function $f$. Here we provide some examples:

1. i)

By considering the function $f(z)=\cos(\lambda z)$ with $\lambda\in\mathbb{C}$, one gets

[TABLE]

which for $z=0$ reduces to $\sum\limits_{\gamma\in\Gamma}\chi(\gamma)\cos(\lambda\gamma)e^{-\frac{\nu}{2}|\gamma|^{2}}=0.$ 2. ii)

Other interesting identities involve even polynomials. Indeed, for $f(z)=z^{2p}$, $p=0,1,2,\cdots$, we obtain

[TABLE]

In particular $\sum\limits_{\gamma\in\Gamma}\chi(\gamma)\gamma^{2p}e^{-\frac{\nu}{2}|\gamma|^{2}}=0.$ More generally, for every positive integer $k=0,1,2,\cdots,$ we have

[TABLE]

Indeed, the odd case, i.e., $k=2p+1$ in (3.12), can be handled using the change of the summation index taking into account that $\chi$ is even. In fact this yields $\psi_{\Gamma}(k)=(-1)^{k}\psi_{\Gamma}(k)$, so that $\psi_{\Gamma}(2p+1)=0$.

Remark 3.9**.**

The obtained identities (3.12), are exactly those obtained by Perelomov [10, Equation (47)], for regular lattice with cell area $S=\pi$ (so $\nu=1$), when dealing with the completeness of the coherent state system. The proof of the simple case $p=0,$ given there requires a detailed knowledge of the relationships between theta functions.

In the next section, we proceed towards a proof of our main result.

4. Proof of main results (Theorem 1.2):

Proof of i): We have to prove the following representation series for the Weierstrass $\sigma$-function

[TABLE]

with ${\mathscr{H}}_{o}(\Gamma)=\nu\sum_{\gamma\in\Gamma}\chi(\gamma)|\gamma|^{2}e^{-\frac{\nu}{2}|\gamma|^{2}}$. Indeed, since $\widetilde{\theta_{{}_{W}}}(z;\Gamma)$ and $\widetilde{\sigma}(z;\Gamma)$ are both generators of the one dimensional space ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$ (see Theorem 3.5 and Theorem 3.2), there exists a constant $C\neq 0$ such that $\widetilde{\theta_{{}_{W}}}(z;\Gamma)=C\widetilde{\sigma}(z;\Gamma)$. This reads explicitly as

[TABLE]

Differentiation of both sides of (4.2) at $z=0$ yields

[TABLE]

This leads to ${\mathscr{H}}_{o}(\Gamma)=C\neq 0$, since the left hand side in (4.3) is exactly ${\mathscr{H}}_{o}(\Gamma)$. Using this fact in (4.2) gives rise to the representation series (4.1).

Proof of ii): According to Theorem 3.2, the modified Weierstrass $\sigma$-function $\widetilde{\sigma}(z;\Gamma):=e^{-\frac{1}{2}\mu z^{2}}\sigma(z;\Gamma)$ belongs to ${\mathcal{O}}^{\nu}_{\Gamma,\chi}(\mathbb{C})$. By iii) of Theorem 2.4, we get

[TABLE]

and therefore, we obtain the reproducing integral formula over $\mathbb{C}$ for the Weierstrass $\sigma$-function

[TABLE]

This completes the proof of ii).

Proof of iii): The expression for the Taylor coefficients of $\sigma(z;\Gamma)$ can be easily obtained by making use of the obtained representation series (4.1) combined with the following technical lemma.

Lemma 4.1**.**

For arbitrary complex numbers $a$ and $b$, we have the following expansion in power series of the exponential function $e^{az^{2}+bz}:$

[TABLE]

where ${\mathbf{F}}(a;c;x)$ is the usual confluent hypergeometric function.

In fact, substitution of (4.6) into the representation series (4.1), with $a=\mu/2$ and $b=\nu\overline{\gamma}$, gives

[TABLE]

The change of $\gamma$ by $-\gamma$ in the first summation of the right hand side and the use of the fact $\chi(-\gamma)=\chi(\gamma)$ give rise to

[TABLE]

Therefore the above expression for $\sigma(z;\Gamma)$ reduces further to the following one

[TABLE]

where we have set

[TABLE]

This completes the proof of iii) of Theorem 1.2.

Proof of iv): This is an immediate consequence of the reproducing integral formula (4.5) (i.e., ii) of Theorem 1.2). Indeed, by replacing the involved exponential function $e^{\frac{\mu}{2}z^{2}+\nu z\overline{w}}$ by its expansion in power series as given by (4.6) (with $a=\mu/2$ and $b=\nu\overline{w}$), and comparing the obtained expression with

[TABLE]

one easily deduces the representation of the coefficients ${\mathscr{W}}_{r}(\Gamma)$ in terms of the Gauss-Hermite integrals.

This completes the proof of Theorem 1.2. $\square$

Remark 4.2**.**

The technical Lemma 4.1 is basically an alternative appropriate form of $e^{-z^{2}+2tz}$ which is the generating function for the Hermite polynomials $H_{n}(t)$. Indeed, splitting the obtained sum by collecting terms in $z^{2r}$ and those in $z^{2r+1}$ and then applying the transformations [8, page 252]

[TABLE]

and

[TABLE]

give rise to the result of Lemma 4.1.

Acknowledgements

The authors are thankful to the anonymous referee and to the editor for their valuable suggestions for improving the presentation of the paper.