Evaluation of some sums involving powers of harmonic numbers

Ce Xu (), Xixi Zhang () and Jianqiang Zhao ()
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Ce Xu: Anhui Normal University
Xixi Zhang: Lixin No.1 Middle School of Anhui
Jianqiang Zhao: The Bishop’s School

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 1, 357-366

Abstract: Abstract In this note, we extend the definition of multiple harmonic sums and apply their stuffle relations to obtain explicit evaluations of the sums $$R_n(p,t)=\sum \nolimits _{m=0}^n m^p H_m^t$$ R n ( p , t ) = ∑ m = 0 n m p H m t for $$t = 2, 3, 4$$ t = 2 , 3 , 4 , where $$H_m$$ H m are harmonic numbers. When $$t\le 4$$ t ≤ 4 these sums were first studied by Spieß 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 $$R_n(p,t)$$ R n ( p , t ) for all $$t\ge 0$$ t ≥ 0 .

Keywords: Bernoulli number; Harmonic number; Multiple harmonic sum; Extended multiple harmonic sum; 05A19; 11B73; 11M32; 68R05 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00486-z

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