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arXiv · 2107.06864

Evaluation of Some Sums Involving Powers of Harmonic Numbers

Ce Xu·Xixi Zhang·Jianqiang Zhao
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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 $t\ge 0$.

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Ce Xu, Xixi Zhang, Jianqiang Zhao. 2021-06-25. Evaluation of Some Sums Involving Powers of Harmonic Numbers. https://arxiv.org/abs/2107.06864

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