Evaluation of Euler-Zagier sums (Q1348717)

A short and elegant evaluation of the triple sums \[ w(p,q,r):= \sum_{n,m=1}^{\infty} {1 \over n^p m^q (n+m)^r} \] in terms of single and double zeta values is given, where \(p>1\), \(r\geq 1\) are integers and \(q\geq 0\) is real with \(q+r>1\). The proof is based on a partial fraction identity and subsequent evaluation for the sum \(\sum_{n=1}^\infty {1 \over n^p (n+x)}\). This extends results going back to \textit{L. Tornheim} [Am. J. Math. 72, 303-314 (1950; Zbl 0036.17203)].