A note on some identities of degenerate Bernoulli numbers

Taekyun Kim, Dae San Kim, Hyunseok Lee, and Misbah Khan
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 3, Pages 536–543
DOI: 10.7546/nntdm.2026.32.3.536-543
Full paper (PDF, 173 Kb)

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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

Hyunseok Lee
Department of Mathematics, Kwangwoon University
Seoul 139-701, Republic of Korea

Misbah Khan
Department of Mathematics, Kwangwoon University
Seoul 139-701, Republic of Korea

Abstract

In this note, we investigate the algebraic and combinatorial properties of degenerate special numbers and polynomials, focusing on their deep connections to classical sequences. By leveraging the framework of degenerate exponential and logarithm functions, we establish several explicit formulas and identities. Specifically, we provide an explicit expression for the Bessel numbers of the first kind and recall a previously established explicit formula for the degenerate Bernoulli numbers that is needed for the subsequent development. Finally, we establish an identity linking the Bessel numbers of the first kind, the degenerate Bernoulli numbers, and the degenerate Stirling numbers of the first kind.

Keywords

• Degenerate Bernoulli numbers
• Bessel numbers of the first kind
• Degenerate Stirling numbers of the first kind

2020 Mathematics Subject Classification

• 11B68
• 11B73
• 11B83

References

1. Benmoussa, A. (2025). A new identity linking Bernoulli numbers, Stirling numbers of the first kind, and Bessel numbers of the first kind. Preprint. Available online at: https://arxiv.org/abs/2505.22819.
2. Carlitz, L. (1979). Degenerate Stirling, Bernoulli and Eulerian numbers. Utilitas Mathematica, 15, 51–88.
3. Comtet, L. (1974). Advanced Combinatorics. The Art of Finite and Infinite Expansions (revised and enlarged edition). D. Reidel Publishing Company, Dordrecht.
4. Kim, D. S., & Kim, T. (2026). Probabilistic degenerate logarithm and heterogeneous Stirling numbers. Kuwait Journal of Science, 53(3), Article ID 100579.
5. Kim, T., & Kim, D. S. (2025). Heterogeneous Stirling numbers and heterogeneous Bell polynomials. Russian Journal of Mathematical Physics, 32(3), 498–509.
6. Kim, T., & Kim, D. S. (2025). Recurrence relations for degenerate Bell and Dowling polynomials via Boson operators. Computational Mathematics and Mathematical Physics, 65(9), 2087–2096.
7. Kim, T., & Kim, D. S. (2025). Spivey-type recurrence relations for degenerate Bell and Dowling polynomials. Russian Journal of Mathematical Physics, 32(2), 288–296.
8. Kim, T., & Kim, D. S. (2026). Degenerate algorithms for degenerate Bernoulli and Euler numbers. Georgian Mathematical Journal. Available online at: https://doi.org/10.1515/gmj-2026-3022.
9. Kim, T., & Kim, D. S. (2026). Degenerate Euler–Seidel matrix method and their applications. Mathematical Methods in the Applied Sciences, 49(8), 8209–8223.
10. Kim, T., Kim, D. S., Lee, H., & Hwang, K.-S. (2026). Degenerate Euler-Seidel method for degenerate Bernoulli, Euler, and Genocchi polynomials. Networks and Heterogeneous Media, 21(2), 551–563.
11. Roman, S. (1984). The Umbral Calculus. Pure and Applied Mathematics, Vol. 111, Academic Press, Inc., New York.
12. Stenlund, D. (2022). On the connection between Stirling numbers and Bessel numbers. The Electronic Journal of Combinatorics, 29(1), Article ID P1.40.

Manuscript history

• Received: 25 June 2026
• Revised: 20 October 2026
• Accepted: 21 October 2026
• Online First: 21 October 2026

Copyright information

Ⓒ 2026 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).