Rapidly convergent series representations of symmetric Tornheim double zeta functions

Abstract

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 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 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 $$d_{kr} \in {\mathbb{Z}}_{\ge 0}$$ .