Abstract
We give closed-form evaluation formulas for the real and imaginary parts of
the series $\sum_{m,n=1}^{\infty}\frac{e^{2\pi i\left( mx-ny\right) }}
{m^{p}n^{r}\left( mc+n\right) ^{q}},$ $c\in\mathbb{N},$ in terms of certain
zeta values.\textbf{ }Particular choices of $x$ and $y$ lead evaluation
formulas for some Tornheim type $\sum_{m,n=1}^{\infty}\frac{1}{m^{p}%
n^{r}\left( mc+n\right) ^{q}}$ and Euler type $\sum_{m,n=1}^{\infty}\frac
{1}{n^{p}\left( mc+n\right) ^{q}}$ double series and their alternating analogues.
Subject
Geometry and Topology,Statistics and Probability,Algebra and Number Theory,Analysis