Rapidly convergent series representations of symmetric Tornheim double zeta functions

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 functions. As a corollary, give new proof known results on values s, s) at non-positive integers location poles s). Furthermore, prove that can not be written by polynomial in form $$\sum_{k=1}^j c_k \prod_{r=1}^q \zeta^{d_{kr}} (a_{kr} s + b_{kr})$$ where $$a_{kr}, b_{kr}, $$d_{kr} {\mathbb{Z}}_{\ge 0}$$ .