On the evaluation of Tornheim sums and allied double sums

Abstract

The object of study in this paper is some Tornheim type sums $\sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \frac{1}{n^{r}m^{s}(n+m)^{t}}$ which are close relatives of the so-called Euler sums $\sum_{n=1}^{\infty}\frac{1}{n^{s}}\sum_{m=1}^{\infty}\frac{1}{m^{t}}$. Closed form evaluations of several such double sums are obtained using elementary summation techniques earlier developed by the same author.