Rapidly convergent series representations of symmetric Tornheim double 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 $d_{kr} \in {\mathbb{Z}}_{\ge 0}$.