A note on normal ordering of degenerate integral powers of number operator

Taekyun Kim, Dae San Kim and Kyo-Shin Hwang
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 4, Pages 785–791
DOI: 10.7546/nntdm.2025.31.4.785-791
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Authors and affiliations

Taekyun Kim
Department of Mathematics, Kwangwoon University
Seoul 139-701, Republic of Korea

Dae San Kim
Department of Mathematics, Sogang University
Seoul 121-742, Republic of Korea

Kyo-Shin Hwang
Graduate School of Education, Yeungnam University
Gyeongsan 3854, Republic of Korea

Abstract

This study derives the normal ordering expansion of degenerate integral powers of the number operator, (a^{\dagger}a)_{n,\lambda}, using recurrence relations for the coefficients and an operator action on number states. Here a^{\dagger} and a are respectively the boson creation and annihilation operators. We also determine the inverse of this normal ordering expansion. By analyzing diagonal coherent state elements of the degenerate integral powers of the number operator, we establish a combinatorial identity which yields a Dobinski-like formula for the degenerate Bell numbers at a specific value, connecting degenerate quantum operator calculus with combinatorics.

Keywords

  • Degenerate integral powers
  • Normal ordering
  • Degenerate Stirling numbers of the second kind

2020 Mathematics Subject Classification

  • 11B73
  • 11B83

References

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Manuscript history

  • Received: 14 September 2025
  • Revised: 25 October 2025
  • Accepted: 27 November 2025
  • Online First: 5 November 2025

Copyright information

Ⓒ 2025 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

Kim, T., Kim, D. S., & Hwang, K.-S. (2025). A note on normal ordering of degenerate integral powers of number operator. Notes on Number Theory and Discrete Mathematics, 31(4), 785-791, DOI: 10.7546/nntdm.2025.31.4.785-791.

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