Quotients of sequences under the binomial convolution

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 \{a(n)\} of the equation

    \begin{equation*} \sum_{k=0}^n{n \choose k}a(k)b(n-k)=c(n), \ n=0,1,2,\ldots, \end{equation*}

where b(0)\ne 0, that is, of the equation a \circ b = c in a, where \circ 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

References

  1. Aigner, M. (1979). Combinatorial Theory. Springer-Verlag, New York.
  2. Alecci, G., Barbero, S., & Murru, N. (2023). Some notes on the algebraic structure of linear recurrent sequences. Ricerche di Matematica, 1–17.
  3. 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.
  4. Gessel, I. M., & Kar, I. (2024). Binomial convolutions for rational power series. Journal of Integer Sequences, 27(1), Article 24.1.3.
  5. Graham, R. L., Knuth, D. E., & Patashnik, O. (1989). Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley Publishing Company, New York.
  6. Haukkanen, P. (1994). Roots of sequences under convolutions. The Fibonacci Quarterly, 32, 369–372.
  7. Haukkanen, P. (1996). On a binomial convolution of arithmetical functions. Nieuw Archief voor Wiskunde, 14, 209–216.
  8. Haukkanen, P. (2023). Quotients of arithmetical functions under the Dirichlet convolution. Notes on Number Theory and Discrete Mathematics, 29(2), 185–194.
  9. Kılıç, E., Akkuş, I., Ömür, N., & Ulutaş, Y. T. (2016). Generalized binomial convolution of the mth powers of the consecutive integers with the general Fibonacci sequence. Demonstratio Mathematica, 49(4), 379–385.
  10. Pellegrino, F. (1963). La divisione integrale. Rendiconti di Matematica e delle sue
    Applicazioni, 22(5), 489–497.
  11. Riordan, J. (1958). An Introduction to Combinatorial Analysis. John Wiley and Sons, New York.
  12. Schinzel, A. (1998). A property of the unitary convolution. Colloquium Mathematicae, 78(1), 93–96.
  13. Singh, J. (2014). On an arithmetic convolution. Journal of Integer Sequences, 17(2), Article 14.6.7.
  14. Srisopha, S., Ruengsinsub, P., & Pabhapote, N. (2012). The Qα–convolution of arithmetic functions and some of its properties. East–West Journal of Mathematics, 14(01), 76–83.
  15. Succi, F. (1956). Sulla espressione del quoziente integrale di due funzioni aritmetiche. Rendiconti di Matematica e delle sue Applicazioni, 15(5), 80–92.

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).

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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.

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