Evaluation of Some Sums Involving Powers of Harmonic Numbers

TL;DR

This paper extends the definition of multiple harmonic sums, derives explicit formulas for sums involving powers of harmonic numbers, and provides a general structural understanding of these sums for all non-negative powers.

Contribution

It introduces an extended definition of multiple harmonic sums and derives explicit evaluations for sums involving powers of harmonic numbers, generalizing previous results.

Findings

Explicit formulas for sums R_n(p,t) involving harmonic numbers.

Structural results valid for all non-negative powers t.

Extension of multiple harmonic sums definition.

Abstract

In this note, we extend the definition of multiple harmonic sums and apply their stuffle relations to obtain explicit evaluations of the sums $R_n(p,t)=\sum\nolimits_{m=0}^n m^p H_m^t$, where $H_m$ are harmonic numbers. When $t\le 4$ these sums were first studied by Spie\ss\ around 1990 and, more recently, by Jin and Sun. Our key step first is to find an explicit formula of a special type of the extended multiple harmonic sums. This also enables us to provide a general structural result of the sums $R_n(p,t)$ for all $t\ge 0$.