Generalized Harmonic Number Sums and Quasi-Symmetric Functions

Abstract

We express some general type of infinite series such as $$\sum _{n=1}^{\infty }\frac{F(H_n^{(m)}(z),H_n^{(2m)}(z),\dots ,H_n^{(\ell m)}(z))}{(n+z{)}^{s_1}(n+1+z{)}^{s_2}\cdots (n+k-1+z{)}^{s_k}},$$ where $F(x_1,\dots ,x_{\ell })\in \mathbb{Q}[x_1,\dots ,x_{\ell }]$, $H_n^{(m)}(z)={\sum }_{j=1}^{n}1/(j+z{)}^m$, $z\in (-1,0]$, 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 speical values of $H_n^{(m)}(z)$.

22 pages

References in corpus (2)

• Harmonic-Number Summation Identities, Symmetric Functions, and Multiple Zeta Values
• On some combinatorial identities and harmonic sums