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 $\mathbb{C}$.