Series associated with harmonic numbers, Fibonacci numbers and central binomial coefficients \binom{2n}{n}

Segun Olofin Akerele and Olamide Esther Salami
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 1, Pages 82–90
DOI: 10.7546/nntdm.2025.31.1.82-90
Full paper (PDF, 265 Kb)

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Authors and affiliations

Segun Olofin Akerele
Department of Mathematics, University of Ibadan
Ibadan, Oyo-State, Nigeria

Olamide Esther Salami
Department of Mathematics, University of Ibadan
Ibadan, Oyo-State, Nigeria

Abstract

We find various series that involve the central binomial coefficients \binom{2n}{n}, harmonic numbers and Fibonacci numbers. Contrary to the traditional hypergeometric function _pF_q approach, our method utilizes a straightforward transformation to obtain new evaluations linked to Fibonacci numbers and the golden ratio. We also gave a new series representation for \zeta(2).

Keywords

  • Central binomial coefficients
  • Harmonic number
  • Catalan number
  • Fibonacci number
  • Lucas number

2020 Mathematics Subject Classification

  • 40A05
  • 11B39
  • 11B65
  • 05A10

References

  1. Abramowitz, M., & Stegun, I. A. (1965). Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Vol. 55). Courier Corporation.
  2. Adegoke, K., Frontczak, R., & Goy, T. (2024). On some series involving the binomial coefficients \binom{3n}{n}. Notes on Number Theory and Discrete Mathematics, 30(2), 319–334.
  3. Bhandari, N. (2022). Binomial sums resulting to the ratio, G/\pi and 1/\pi. Archimede Mathematical Journal, 9(1), 4–11.
  4. Chen, H. (2016). Interesting series Associated with Central Binomial Coefficients, Catalan Numbers and Harmonic Numbers. Journal of Integer Sequences, 19(1), Article 16.1.5.
  5. Chen, H. (2022). Interesting Ramanujan-like series associated with powers of central binomial coefficients. Journal of Integer Sequences, 25(1), Article 22.1.8.
  6. Graham, R. L., Knuth, D. E., & Patashnik, O. (1994). Concrete Mathematics. (2nd ed.). Addison-Wesley Publishing Company, Inc.

Manuscript history

  • Received: 14 June 2024
  • Revised: 2 April 2025
  • Accepted: 7 April 2025
  • Online First: 7 April 2025

Copyright information

Ⓒ 2025 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

Akerele, S. O., & Salami, O. E. (2025). Series associated with harmonic numbers, Fibonacci numbers and central binomial coefficients \binom{2n}{n}. Notes on Number Theory and Discrete Mathematics, 31(1), 82-90, DOI: 10.7546/nntdm.2025.31.1.82-90.

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