  We express some general type of infinite series such as $$
\sum^\infty_{n=1}\frac{F(H_n^{(m)}(z),H_n^{(2m)}(z),\ldots,H_n^{(\ell m)}(z))}
{(n+z)^{s_1}(n+1+z)^{s_2}\cdots (n+k-1+z)^{s_k}}, $$ where
$F(x_1,\ldots,x_\ell)\in\mathbb Q[x_1,\ldots,x_\ell]$,
$H_n^{(m)}(z)=\sum^n_{j=1}1/(j+z)^m$, $z\in (-1,0]$, and $s_1,\ldots,s_k$ are
nonnegative integers with $s_1+\cdots+s_k\geq 2$, as a linear combination of
multiple Hurwitz zeta functions and some speical values of $H_n^{(m)}(z)$.
