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 $d_{ kr} \in { \mathbb{ Z}}_{ \ge 0}$.
Categories: math.NT