A functional equation for multiple zeta functions and generalized confluent hypergeometric functions

TL;DR

The paper addresses the problem of a higher-dimensional functional equation for Euler–Zagier multiple zeta functions by introducing a new framework based on multiple confluent hypergeometric functions. It constructs the objects $\mathscr{F}_\pm^r$ and $\mathscr{G}_r$ via nested integrals involving the generalized hypergeometric forms and uses contour methods to derive a functional equation that generalizes Matsumoto's when $r=2$. The results include meromorphic continuation of $\mathscr{F}_\pm^r$ and $\mathscr{G}_r$ to the full spaces $\mathfrak{A}_r$ and $\mathbb{C}^r$, and connections to Lauricella functions and zeta functions of root systems. This work provides a unified approach to higher-rank multiple zeta phenomena and links to Mordell–Tornheim-type structures.

Abstract

In this paper, we introduce a new function, the multiple confluent hypergeometric functions, and establish a functional equation for the $r$-variable Euler--Zagier multiple zeta functions using it. In the case when $r=2$, this functional equation includes the well-known functional equation for the Euler--Zagier double zeta functions obtained by Matsumoto.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2: Mat1
• Theorem 1.3: Oka and Ono
• Definition 1.4: Multiple confluent hypergeometric functions
• Theorem 1.5
• Remark 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof