Generalized hyperharmonic number sums with reciprocal binomial coefficients
type: Item
instance of: scholarly article
title: Generalized hyperharmonic number sums with reciprocal binomial coefficients
zbMATH Open document ID: 1505.05008
author: Rusen Li
published in: Mathematica Slovaca
publication date: 2022-10-19T00:00:00Z
review text: The classical harmonic numbers are the sums \[ H_n^{(p)} = \sum_{j=1}^n \frac{1}{j^p}, \] where \(n\) and \(p\) are positive integers. The generalized hyperharmonic numbers are defined by the induction \[ H_n^{(p,r)} = \sum_{j=1}^n H_j^{(p,r-1)} \] and \(H_n^{(p,1)}=H_n^{(p)}\), where \(n\), \(p\), and \(r\) are positive integers. The main goal of this article is to give exact formulas for the sums \[ \sum_{n=1}^{\infty} \frac{H_n{(p,s)}}{n^m \binom{n+k}{k}}, \qquad \sum_{n=1}^{\infty} (-1)^{n+1} \frac{H_n{(p,s)}}{n^m \binom{n+k}{k}}, \] and \[ \sum_{n=1}^{\infty} \frac{H_n{(p_1,s_1)}H_n{(p_2,s_2)}}{n^m \binom{n+k}{k}}, \qquad \sum_{n=1}^{\infty} (-1)^{n+1} \frac{H_n{(p_1,s_1)}H_n{(p_2,s_2)}}{n^m \binom{n+k}{k}}. \] These formulas are given in terms of auxiliary quantities \(a(s,\ell_1,\ell_2)\), \(S(p,m,t,r,x)\) and \(T(p_1,p_2,m,t,r,x)\). In the examples, it is shown that these formulas can be given in terms of zeta values, powers of \(\pi\), polylogarithms, etc.
reviewed by: Matthieu Josuat-Vergés
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 7604032
zbMATH Keywords: generalized hyperharmonic numbers
zbMATH Keywords: classical Euler sums
zbMATH Keywords: binomial coefficients
zbMATH Keywords: combinatorial approach
zbMATH Keywords: partial fraction approach
MaRDI profile type: Publication
full work available at URL: ms-2022-0076
OpenAlex ID: W4312373184
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DOI: 10.1515/MS-2022-0076
description: scientific article; zbMATH DE number 7604032