Multiple $\wp$-Functions and Their Applications

TL;DR

This paper introduces multiple $ ext{wp}$-functions generalizing the classical Weierstrass $ ext{wp}$-function, establishing explicit formulas and exploring their applications to relations among Eisenstein series and zeta values.

Contribution

It presents the definition of multiple $ ext{wp}$-functions, derives explicit formulas relating them to single $ ext{wp}$-functions, and applies these to study Eisenstein series and zeta values.

Findings

Explicit formulas expressing multiple $ ext{wp}$-functions in terms of single $ ext{wp}$-functions.

Relations among multiple Eisenstein series and multiple zeta values derived.

Double periodicity of multiple $ ext{wp}$-functions exploited for applications.

Abstract

In this paper, we introduce and study multiple $\wp$-functions, which generalize the classical Weierstrass $\wp$-function to iterated sums over lattice points, and we establish explicit formulas expressing them in terms of single $\wp$-functions with coefficients given by multiple Eisenstein series. As an application, we derive some relations among multiple Eisenstein series and multiple zeta values by exploiting the double periodicity of the multiple $\wp$-functions.