Powers of the operator u(z)\frac{d}{dz} and their connection with some combinatorial numbers

Ioana Petkova
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 4, Pages 716–734
DOI: 10.7546/nntdm.2024.30.4.716-734
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Authors and affiliations

Ioana Petkova
Faculty of Mathematics and Informatics, Sofia University
Sofia, Bulgaria

Abstract

In this paper the operator A = u(z)\frac{d}{dz} is considered, where u is an entire or meromorphic function in the complex plane. The expansion of A^{k} (k\geq1) with the help of the powers of the differential operator D=\frac{d}{dz} is obtained, and it is shown that this expansion depends on special numbers. Connections between these numbers and known combinatorial numbers are given. Some special cases of the operator A, corresponding to u(z) = z, u(z) = e^{z}, u(z) = \frac{1}{z}, are considered.

Keywords

  • Combinatorial numbers
  • Special numbers
  • Operators
  • Euler–Cauchy operator
  • Bessel functions
  • Entire functions
  • Meromorphic functions

2020 Mathematics Subject Classification

  • 33C10
  • 11B73
  • 11F25
  • 30D10

References

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Manuscript history

  • Received: 16 July 2024
  • Accepted: 2 October 2024
  • Online First: 8 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

Petkova, I. (2024). Powers of the operator u(z)\frac{d}{dz} and their connection with some combinatorial numbers. Notes on Number Theory and Discrete Mathematics, 30(4), 716-734, DOI: 10.7546/nntdm.2024.30.4.716-734.

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