On generalized hyperharmonic numbers of order r, H_{n,m}^{r} (\sigma)

Sibel Koparal, Neşe Ömür and Laid Elkhiri
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 4, Pages 804–812
DOI: 10.7546/nntdm.2023.29.4.804-812
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Authors and affiliations

Sibel Koparal
Department of Mathematics, University of Bursa Uludağ
16059 Nilufer, Bursa, Turkey

Neşe Ömür
Department of Mathematics, University of Kocaeli
41380 Izmit, Kocaeli, Turkey

Laid Elkhiri
Faculty of Material and Sciences, University of Tiaret
Algeria

Abstract

In this paper, we define generalized hyperharmonic numbers of order r, H_{n,m}^{r}\left( \sigma \right) , for m\in \mathbb{Z}^{+} and give some applications by using generating functions of these
numbers. For example, for n, r, s\in \mathbb{Z}^{+} such that 1\leq s\leq r,

    \begin{equation*} \sum\limits_{k=1}^{n}\binom{n-k+s-1}{s-1}H_{k,m}^{r-s}\left( \sigma \right) =H_{n,m}^{r}\left( \sigma \right), \end{equation*}

and

    \begin{equation*} \sum_{k=1}^{n}\sum_{i=1}^{k}\frac{H_{k-i,m}^{r+1}\left( \sigma \right) D_{r}(k-i+r)}{(n-k)!\left( k-i+r\right) !}=H_{n,m}^{2r+2}(\sigma ), \end{equation*}

where D_{r}(n) is an r-derangement number.

Keywords

  • Sums
  • Generalized harmonic numbers
  • Generating function

2020 Mathematics Subject Classification

  • 05A15
  • 05A19
  • 11B73

References

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Manuscript history

  • Received: 26 February 2023
  • Revised: 13 November 2023
  • Accepted: 23 November 2023
  • Online First: 30 November 2023

Copyright information

Ⓒ 2023 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Koparal, S., Ömür, N., & Elkhiri, L. (2023). On generalized hyperharmonic numbers of order r, H_{n,m}^{r} (\sigma). Notes on Number Theory and Discrete Mathematics, 29(4), 804-812, DOI: 10.7546/nntdm.2023.29.4.804-812.

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