Fibonacci series from power series

Kunle Adegoke
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 3, Pages 44–62
DOI: 10.7546/nntdm.2021.27.3.44-62
Full paper (PDF, 214 Kb)

Details

Authors and affiliations

Kunle Adegoke
Department of Physics, Obafemi Awolowo University
220005 Ile-Ife, Nigeria

Abstract

We show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For illustrative purposes, Fibonacci series arising from trigonometric functions, the gamma function and the digamma function are derived. Infinite series involving Fibonacci and Bernoulli numbers and Fibonacci and Euler numbers are also obtained.

Keywords

• Fibonacci number
• Lucas number
• Summation identity
• Series
• Generating function
• Gamma function
• Digamma function
• Trigonometric functions
• Bernoulli number
• Zeta function

2020 Mathematics Subject Classification

• 11B39
• 11B37

References

1. Edwards, H. M. (1974). Riemann’s Zeta Function. Academic Press.
2. Erdelyi, A., Magnus, W., Oberhettinger, F., & Tricomi, F. G. (1981). Higher Transcendental Functions, Vol. 1, Bateman manuscript project.
3. Frontczak, R. (2019). Relating Fibonacci numbers to Bernoulli numbers via
   balancing polynomials. Journal of Integer Sequences, 22, Article 18.5.3.
4. Frontczak, R. (2020). Infinite series involving Fibonacci numbers and the Riemann zeta function. Notes on Number Theory and Discrete Mathematics, 26(2), 159–166.
5. Frontczak, R. (2020). Problem B-1267, The Fibonacci Quarterly, 58(2).
6. Frontczak, R. (2020). Problem H-859, The Fibonacci Quarterly, 58(3).
7. Frontczak, R., & Goy, T. (2020). General infinite series evaluations involving Fibonacci  numbers and the Riemann zeta function. Matematychni Studii, 55(2), 115–123.
8. Frontczak, R., & Goy, T. (2020). More Fibonacci–Bernoulli relations with and
   without balancing polynomials. Preprint. Available online at: https://arxiv.org/abs/2007.14618.
9. Gradshteyn, I. S., & Ryzhik, I. M. (2007). Table of Integrals, Series and Products. Academic  Press.
10. Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications. Wiley-Interscience.
11. Riordan, J. (1968). Combinatorial Identities, John Wiley & Sons, Inc. New York–London–Sydney.
12. Srivastava, H. M., & Choi, J. (2012). Zeta and q-Zeta Functions and Associated Series and  Integrals, Elsevier Inc.
13. Vajda, S. (2008). Fibonacci and Lucas Numbers, and the Golden Section: Theory
    and Applications, Dover Press.