Taekyun Kim and Dae San Kim
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 2, Pages 263–268
DOI: 10.7546/nntdm.2026.32.2.263-268
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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
Abstract
In this note, we derive a Leibniz rule for difference quotient.
Keywords
- Leibniz rule
- Difference quotient
2020 Mathematics Subject Classification
- 65Q10
References
- Kim, D. S., & Kim, T. (2021). Degenerate Sheffer sequences and λ-Sheffer sequences. Journal of Mathematical Analysis and Applications, 493(1), Article ID 124521.
- Kim, D. S., & Kim, T. (2025). Combinatorial identities related to degenerate Stirling numbers of the second kind. Proceedings of the Steklov Institute of Mathematics, 330, 176–192.
- Kim, T., & Kim, D. S. (2019). Degenerate Bernstein polynomials. Revista de la Real Academia de Ciencias Exactas, Fısicas y Naturales, Series A Matematicas (RACSAM), 113(3), 2913–2920.
- Kim, T., & Kim, D. S. (2025). Heterogeneous Stirling numbers and heterogeneous Bell polynomials. Russian Journal of Mathematical Physics, 32(3), 498–509.
- 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.
- 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.
- Kim, W. J., Kim, D. S., Kim, H. Y., & Kim, T. (2019). Some identities of degenerate Euler polynomials associated with degenerate Bernstein polynomials. Journal of Inequalities and Applications, 2019, Article ID 160.
- LeVeque, R. J. (2007). Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia.
Manuscript history
- Received: 10 February 2026
- Accepted: 1 April 2026
- Online First: 2 April 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).
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Cite this paper
Kim, T., & Kim, D. S. (2026). A note on Leibniz rule for difference quotient. Notes on Number Theory and Discrete Mathematics, 32(2), 263-268, DOI: 10.7546/nntdm.2026.32.2.263-268.
