Motivated by analytic continuation of solutions of a Picard-Fuchs equation, we encountered sums of the following form

$S(z;p)=\sum_{k=1}^{\infty}(-1)^{k+1} (H_k)^p z^k$

where $H_k = \sum_{n=1}^{k} 1/n$ are the harmonic numbers and $p \in \mathbb{N}.$

For $p=1$, $S(z;1) = \frac{\log(1+z)}{1+z}$ and at $z=1$, $S(z=1;1)=1/2 \log 2$. In general, we expect that the sums for $p>1$ will have a radius of convergence 1, with a pole at $z=-1$, but that the analytic continuation to $z=1$ will have a finite value. This value is expected to be a sum of multiple zeta functions. Are these sums known for $p=2,3,\dots?$

p-th order partial derivatives of the polylogarithmic and Lerch $\Phi$ functions. $\endgroup$ – Lucian Apr 1 '14 at 13:25