On some infinite series of L. J. Mordell and their analogues (Q1063061)

The authors prove a reciprocity relation for the double series \[ S_{f,g}(u,v;w)=\sum^{\infty}_{r,k=1}\frac{f(r+k) g(k)}{r^ u k^ v (r+k)^ w} \] where \(f,g\) are bounded maps from \(\mathbb N\) in \(\mathbb C\), \(u, v\) are positive integers and \(w>0\) is real. From this they deduce \[ \sum^{\infty}_{r,k=1}(-1)^{r+k}/rk(r+k)=(1/4)\zeta (3),\quad \sum^{\infty}_{r,k=1}(-1)^{k-1}/rk(r+k)=(5/8)\zeta (3) \] and \[ \sum^{\infty}_{r,k=1}1/rk(r+k)^ w=(w+1) \zeta (w+2)- \sum^{w}_{i=1}\zeta (i) \zeta (w+2-i) \] where \(w\in \mathbb N\) and \(\zeta\) denotes the Riemann zeta-function. Furthermore from a reciprocity law for \(M(m,n,p)=\sum^{\infty}_{r,k=1}1/r^ m k^ n (r+k)^ p\) they compute \(M(2m,2m,2m)\). Some related multiple series are calculated, too.