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Multiple $\wp$-Functions and Their Applications

Hayato Kanno, Katsumi Kina

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.

Multiple $\wp$-Functions and Their Applications

Abstract

In this paper, we introduce and study multiple -functions, which generalize the classical Weierstrass -function to iterated sums over lattice points, and we establish explicit formulas expressing them in terms of single -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 -functions.

Paper Structure

This paper contains 10 sections, 29 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Analytic properties of multiple $\wp$-functions
  3. Elliptic functions
  4. Basic properties of multiple $\wp$-functions
  5. Explicit formula for the multiple $\wp$-function with repeated index
  6. Multiple $\wp$-function with general index
  7. The algebra of multiple $\wp$-functions
  8. Relations among multiple Eisenstein series from multiple $\wp$-functions
  9. The algebra of multiple $\wp$-functions
  10. Some properties of $\wp_{2,\dots,2}(z;\tau)$

Key Result

Theorem 1.1

For integers $k_1,\dots,k_r\geq 2$ with $k_1+\cdots+k_r=k$, we have $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (59)

  • Theorem 1.1: \ref{['thm:depth-2-wp-formula']}
  • Lemma 2.1: see Lang
  • Definition
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 49 more