On an analytical study of the generalized Fibonacci polynomials

Leandro Rocha, Gabriel F. Pinheiro and Elen V. P. Spreafico
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 3, Pages 535–546
DOI: 10.7546/nntdm.2025.31.3.535-546
Full paper (PDF, 1264 Kb)

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Authors and affiliations

Leandro Rocha
Faculty of Computing, Federal University of Mato Grosso do Sul – UFMS
Cidade Universitária, ave. Costa e Silva – Pioneiros – MS, Brazil

Gabriel F. Pinheiro
Institute of Mathematics, Statistics and Scientific Computing, State University of Campinas
Sérgio Buarque de Holanda St., 651, Cidade Universitária – Campinas – SP, Brazil

Elen V. P. Spreafico
Institute of Mathematics, Federal University of Mato Grosso do Sul – UFMS
Cidade Universitária, ave. Costa e Silva – Pioneiros – MS, Brazil

Abstract

In this work, we presented an analytical study of the generalized Fibonacci polynomial of order by using properties of the fundamental system associated with the generalized Fibonacci polynomial. We established the generating function and provided the asymptotic behavior for each system sequence. Moreover, the properties are extended to any generalized Fibonacci type, given the general case’s generating function and asymptotic behavior.

Keywords

• Generalized Fibonacci polynomials
• Fundamental system
• Analytic representations
• Asymptotic behavior
• Generating functions

2020 Mathematics Subject Classification

• 11B39
• 11B75
• 11C20
• 65Q10
• 65Q30

References

1. Alves, F. R. V., & Catarino, P. M. M. C. (2017). A classe dos polinômios bivariados de Fibonacci (PBF): Elementos recentes sobre a evolução de um modelo. Revista Thema, 14(1), 112–136.
2. Ben Taher, R., & Rachidi, M. (2003). On the matrix powers and exponential by the r-generalized Fibonacci sequences methods: The companion matrix case. Linear Algebra and Its Applications, 370, 341–353.
3. Ben Taher, R., & Rachidi, M. (2016). Solving some generalized Vandermonde systems and inverse of their associate matrices via new approaches for the Binet formula. Applied Mathematics and Computation, 290, 267–280.
4. Falcón, S., & Plaza, Á. (2007). The k-Fibonacci sequence and the Pascal 2-triangle. Chaos, Solutions & Fractals, 33(1), 38–49.
5. Florez, R., McAnally, N., & Mukherjee, A. (2017). Identities for the generalized Fibonacci polynomial. ArXiv. Available online at: https://arxiv.org/abs/1702.01855.
6. Hoggatt, V. E. Jr., & Long, C. T. (1974). Divisibility properties of generalized Fibonacci polynomials. The Fibonacci Quarterly, 12(2), 113–120.
7. Koç, A. B., Çakmak, M., Kurnaz, A., & Uslu, K. (2013). A new Fibonacci type collocation procedure for boundary value problems. Advances in Difference Equations, 2013, Article 262.
8. Kurt, A., Yalçınbaş, S., & Sezer, M. (2013). Fibonacci collocation method for solving high-order linear Fredholm integro-differential-difference equations. International Journal of Mathematics and Mathematical Sciences, 2013, Article ID 486013.
9. Meyer, C. D. (2023). Matrix Analysis and Applied Linear Algebra (2nd ed.). Other Titles in Applied Mathematics, Vol. 188. Society for Industrial and Applied Mathematics, Philadelphia.
10. Rocha, L., & Spreafico, E.V.P. (2025). On generalized Fibonacci polynomials. International Journal of Advances in Applied Mathematics and Mechanics (Submitted).
11. Santos, J. P. O., Mello, M. P., & Murari, I. T. C. (2007). Introdução à Análise Combinatória (4th ed.). Editora Ciencia Moderna, Rio de Janeiro. ˆ
12. Spreafico, E. V. P., & Rachidi, M. (2019). Fibonacci fundamental system and generalized Cassini identity. The Fibonacci Quarterly, 57(2), 155–167.
13. Stanley, R. P. (1999). Enumerative Combinatorics: Volume 2. Cambridge Studies in Advanced Mathematics, Vol. 62. Cambridge University Press, Cambridge

Manuscript history

• Received: 8 October 2024
• Revised: 1 August 2025
• Accepted: 4 August 2025
• Online First: 16 August 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).