Moments and asymptotic expansion of derangement polynomials in terms of Touchard polynomials

Mohammad Ghorbani, Mehdi Hassani and Hossein Moshtagh
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 4, Pages 832–842
DOI: 10.7546/nntdm.2024.30.4.832-842
Full paper (PDF, 241 Kb)

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Authors and affiliations

Mohammad Ghorbani
Department of Mathematics, Faculty of Science, University of Zanjan
University Blvd., 45371-38791, Zanjan, Iran

Mehdi Hassani
Department of Mathematics, Faculty of Science, University of Zanjan
University Blvd., 45371-38791, Zanjan, Iran

Hossein Moshtagh
Department of Computer Science, University of Garmsar
35881-15589, Garmsar, Iran

Abstract

In this paper we study the polynomial \widehat{D}_n(x)=n!\sum_{j=0}^n x^j/j!, which is a variant of derangement polynomials. First we obtain an asymptotic expansion for \widehat{D}_n(x) with coefficients in terms of Touchard polynomials. Then, we compute the moments \sum_{n=0}^\infty(\e^xn!-\widehat{D}_n(x))^k for any integer k\geqslant 1 and any real x\in[0,1).

Keywords

  • Derangement polynomial
  • Touchard polynomials
  • Bell numbers

2020 Mathematics Subject Classification

  • 05A05
  • 05A16
  • 11B73

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Manuscript history

  • Received: 12 October 2024
  • Accepted: 26 November 2024
  • Online First: 28 November 2024

Copyright information

Ⓒ 2024 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

Ghorbani, M., Hassani, M., & Moshtagh, H. (2024). Moments and asymptotic expansion of derangement polynomials in terms of Touchard polynomials. Notes on Number Theory and Discrete Mathematics, 30(4), 832-842, DOI: 10.7546/nntdm.2024.30.4.832-842.

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