Rapidly convergent series representations of symmetric Tornheim double zeta functions

TL;DR

This paper introduces rapidly convergent series representations for symmetric Tornheim double zeta functions, providing new proofs of their special values and pole locations, and demonstrating their non-polynomial nature in terms of Riemann zeta functions.

Contribution

It presents new rapidly convergent series formulas for symmetric Tornheim double zeta functions and analyzes their properties, including values, poles, and algebraic structure.

Findings

New series representations for $T(s,t,u)$ and symmetric variants.

Proof of known values of $T(s,s,s)$ at non-positive integers.

Demonstration that $T(s,s,s)$ cannot be expressed as a polynomial in zeta functions.

Abstract

In the present paper, for $s,t,u \in {\mathbb{C}}$, we show rapidly (or globally) convergent series representations of the Tornheim double zeta function $T(s,t,u)$ and (desingularized) symmetric Tornheim double zeta functions. As a corollary, we give a new a proof of known results on the values of $T(s,s,s)$ at non-positive integers and the location of the poles of $T(s,s,s)$. Furthermore, we prove that the function $T(s,s,s)$ can not be written by a polynomial in the form of $\sum_{k=1}^j c_k \prod_{r=1}^q \zeta^{d_{kr}} (a_{kr} s + b_{kr})$, where $a_{kr}, b_{kr}, c_k \in {\mathbb{C}}$ and $d_{kr} \in {\mathbb{Z}}_{\ge 0}$.