Identities for Fibonacci and Lucas numbers

George Grossman, Aklilu Zeleke and Xinyun Zhu
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 4, Pages 670–681
DOI: 10.7546/nntdm.2023.29.4.670-681
Full paper (PDF, 169 Kb)

Details

Authors and affiliations

George Grossman
Department of Mathematics, Central Michigan University
Pearce Hall 217, Mt. Pleasant, MI, USA 48858

Aklilu Zeleke
Department of Statistics and Probability and the Lyman Briggs College,
Michigan State University
East Lansing MI, USA 48824

Xinyun Zhu
Department of Mathematics, University of Texas of the Permian Basin
Odessa, TX, USA 79762

Abstract

In this paper several new identities are given for the Fibonacci and Lucas numbers. This is accomplished by by solving a class of non-homogeneous, linear recurrence relations.

Keywords

• Recurrence relation
• Non-homogeneous
• Fibonacci and Lucas numbers

2020 Mathematics Subject Classification

• 11B39
• 65Q30

References

1. Azarian, M. K. (2012). Fibonacci identities as binomial sums. International Journal of Contemporary Mathematical Sciences, 7(38), 1871–1876.
2. Azarian, M. K. (2012). Identities involving Lucas or Fibonacci and Lucas numbers as binomial sums. International Journal of Contemporary Mathematical Sciences, 7(45), 2221–2227.
3. Grossman, G., Zeleke, A., Zhu, X., & Zdrahal, T. (2016). Recurrence relation with combinatorial identities. International Journal of Pure and Applied Mathematics, 107(4), 939–948.
4. Gulec, H., & Taskara, N. (2009). On the Properties of Fibonacci numbers with
   binomial coefficients. International Journal of Contemporary Mathematical Sciences, 4(25), 1251–1256.
5. Koshy, T. (2001). Fibonacci Numbers and Lucas numbers with Applications. John Wiley & Sons, Inc.
6. Riordan. J. (1968). Combinatorial Identities. John Wiley & Sons. Inc.
7. Singh, P. (1985). The so-called Fibonacci numbers in ancient and medieval India. Historia Mathematica, 12, 229–244.
8. Wilf, H. S. (1994). generatingfunctionology, Academic Press, Inc.
9. Zhang, Y., & Grossman, G. (2016). Polynomial solution for a nonhomogeneous recurrence relation. Pre-print, Oct. 22, 2016. Available online at: https://www.academia.edu/108731005/ .

Manuscript history

• Received: 10 February 2023
• Revised: 18 October 2023
• Accepted: 30 October 2023
• Online First: 1 November 2023

Copyright information

Ⓒ 2023 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).