Generalized harmonic number sums and quasisymmetric functions

Abstract

We express some general type of infinite series such as

$$\sum _{n=1}^{\infty }\frac{F\left(H_n^{\left(m\right)}\left(z\right),H_n^{\left(2m\right)}\left(z\right),\dots ,H_n^{\left(\ell m\right)}\left(z\right)\right)}{{\left(n+z\right)}^{s_1}{\left(n+1+z\right)}^{s_2}\cdots {\left(n+k-1+z\right)}^{s_k}},$$

where $F\left(x_1,\dots ,x_{\ell }\right)\in \mathbb{Q}\left[x_1,\dots ,x_{\ell }\right]$, $H_n^{\left(m\right)}\left(z\right)={\sum }_{j=1}^{n}1∕{\left(j+z\right)}^m$, $z\in \left(-1,0\right]$, and $s_1,\dots ,s_k$ are nonnegative integers with $s_1+\cdots +s_k\ge 2$, as a linear combination of multiple Hurwitz zeta functions and some special values of $H_n^{\left(m\right)}\left(z\right)$.