Four families of summation formulas involving generalized harmonic numbers
type: Item
instance of: scholarly article
title: Four families of summation formulas involving generalized harmonic numbers
zbMATH Open document ID: 1390.11059
author: Jun Wang
author: Chuanan Wei
published in: The Ramanujan Journal
publication date: 2018-02-15T00:00:00Z
review text: The generalized harmonic number means \(H_n^{<\ell>}(x)= \sum_{k=1}^n \frac{1}{(x+k)^{\ell}}.\) For a natural number \(s=0,1,2\), the paper produces closed form expressions for \[ \sum_{k=1}^n k^sH_k^2(x),\quad \sum_{k=1}^n k^sH_k^3(x),\quad \sum_{k=1}^n k^sH_k(x) H_k^{<2>}(x), \] and \[ \sum_{k=1}^n k^sH_k^2(x) H_k^{<2>}(x), \] using Abel's transformation, but the authors would be able to do such calculations for larger \(s\) well. These results generalize a number of published special cases.
reviewed by: László A. Székely
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 6839066
zbMATH Keywords: difference operator
zbMATH Keywords: Abel's transformation
zbMATH Keywords: harmonic numbers
describes a project that uses: SIGMA
MaRDI profile type: Publication
full work available at URL: s11139-016-9871-4
OpenAlex ID: W1876831146
cites work: The Abel-Zeilberger algorithm
cites work: An extension of Zeilberger's fast algorithm to general holonomic functions
cites work: Q4320535
cites work: Hypergéométrie et fonction zêta de Riemann
cites work: Harmonic number identities via hypergeometric series and Bell polynomials
cites work: Computer proofs of a new family of harmonic number identities.
cites work: Symbolic summation assists combinatorics
cites work: Some more identities involving rational sums
cites work: Sums of derivatives of binomial coefficients
cites work: Arithmetic theory of harmonic numbers
cites work: Riordan arrays and harmonic number identities
cites work: Harmonic number identities via the Newton-Andrews method
cites work: A family of summation formulae involving harmonic numbers
DOI: 10.1007/S11139-016-9871-4
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description: scientific article; zbMATH DE number 6839066