# Generalized Harmonic Number Sums and Quasi-Symmetric Functions

## TL;DR

This paper develops a method to express complex infinite series involving generalized harmonic numbers and quasi-symmetric functions as linear combinations of multiple Hurwitz zeta functions and special harmonic number values.

## Contribution

It introduces a novel approach to represent a broad class of infinite series with harmonic number sums in terms of well-known special functions and constants.

## Key findings

- Expressed series as linear combinations of multiple Hurwitz zeta functions.
- Connected harmonic number sums to special values of these functions.
- Provided a framework for evaluating complex series involving harmonic numbers.

## Abstract

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 $H_n^{(m)}(z)$.