Sums of powers of integers and hyperharmonic numbers

José Luis Cereceda
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 2, Pages 101—110
DOI: 10.7546/nntdm.2021.27.2.101-110
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Authors and affiliations

José Luis Cereceda
Collado Villalba, 28400 Madrid, Spain

Abstract

In this paper, we obtain a new formula for the sums of k-th powers of the first n positive integers, Sk(n), that involves the hyperharmonic numbers and the Stirling numbers of the second kind. Then, using an explicit representation for the hyperharmonic numbers, we generalize this formula to the sums of powers of an arbitrary arithmetic progression. Furthermore, we express the Bernoulli polynomials in terms of hyperharmonic polynomials and Stirling numbers of the second kind. Finally, we extend the obtained formula for Sk(n) to negative values of n.

Keywords

  • Sums of powers of integers
  • Hyperharmonic numbers
  • Stirling numbers of the second kind
  • Bernoulli polynomials
  • Hyperharmonic polynomials

2020 Mathematics Subject Classification

  • 11B68
  • 11B25
  • 11B83

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Cite this paper

Cereceda, J. L. (2021). Sums of powers of integers and hyperharmonic numbers. Notes on Number Theory and Discrete Mathematics, 27(2), 101-110, doi: 10.7546/nntdm.2021.27.2.101-110.

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