A functional relation for Tornheim's double zeta functions

TL;DR

This paper extends the theory of Tornheim's double zeta functions by establishing a new relation, providing integral representations, and analyzing their behavior at specific points, thus broadening understanding of multiple zeta values.

Contribution

It introduces a generalized partial fraction decomposition and proves a new relation between Tornheim's double zeta functions of three complex variables.

Findings

New integral representations of zeta functions

Extension of the parity result to the entire domain

Explicit expressions at non-positive integers

Abstract

In this paper, we generalize the partial fraction decomposition which is fundamental in the theory of multiple zeta values, and prove a relation between Tornheim's double zeta functions of three complex variables. As applications, we give new integral representations of several zeta functions, an extension of the parity result to the whole domain of convergence, concrete expressions of Tornheim's double zeta function at non-positive integers and some results for the behavior of a certain Witten's zeta function at each integer. As an appendix, we show a functional equation for Euler's double zeta function.