Pentti Haukkanen
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 4, Pages 681–690
DOI: 10.7546/nntdm.2024.30.4.681-690
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Pentti Haukkanen
Faculty of Information Technology and Communication Sciences,
FI-33014 Tampere University, Finland
Abstract
This paper gives expressions for the solution of the equation
where , that is, of the equation in , where is the binomial convolution. These expressions are classified as recursive, explicit, determinant, exponential generating function and convolutional expressions. These expressions are compared with those under the usual Cauchy convolution. Several special cases and examples of combinatorial nature are also discussed.
Keywords
- Arithmetical equation
- Binomial convolution
- Cauchy convolution
- Exponential generating function
- Binomial coefficient
2020 Mathematics Subject Classification
- 05A10
- 11A25
- 11B65
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Manuscript history
- Received: 15 August 2024
- Revised: 25 October 2024
- Accepted: 26 October 2024
- Online First: 5 November 2024
Copyright information
Ⓒ 2024 by the Author.
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).
Related papers
- Batır, N., & Sofo, A. (2023). Sums involving the binomial coefficients, Bernoulli numbers of the second kind and harmonic numbers. Notes on Number Theory and Discrete Mathematics, 29(1), 78–97.
- Haukkanen, P. (2023). Quotients of arithmetical functions under the Dirichlet convolution. Notes on Number Theory and Discrete Mathematics, 29(2), 185–194.
Cite this paper
Haukkanen, P. (2024). Quotients of sequences under the binomial convolution. Notes on Number Theory and Discrete Mathematics, 30(4), 681-690, DOI: 10.7546/nntdm.2024.30.4.681-690.