An algorithm for complex factorization of the bi-periodic Fibonacci and Lucas polynomials

Baijuan Shi and Can Kızılateş
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 2, Pages 269–279
DOI: 10.7546/nntdm.2025.31.2.269-279
Full paper (PDF, 225 Kb)

Details

Authors and affiliations

Baijuan Shi
School of Science, Xi’an University of Posts and Telecommunications
Xi’an, Shaanxi P. R. China

Can Kızılateş
Department of Mathematics, Faculty of Science, Zonguldak Bulent Ecevit University
Türkiye

Abstract

In this paper, we consider the factorization of generalized sequences, by employing a method based on trigonometric identities. The new method is of reduced complexity and represents an improvement compared to existing results. We establish a connection between the bi-periodic Fibonacci and Lucas polynomials and tridiagonal matrices, which exploits the calculation of eigenvalues of associated tridiagonal matrices.

Keywords

  • Bi-periodic Fibonacci polynomials
  • Bi-periodic Lucas polynomials
  • Tridiagonal matrices
  • Trigonometric identities
  • Eigenvalues
  • Complex factorizations

2020 Mathematics Subject Classification

  • 15A15
  • 15B05
  • 15A60
  • 11B39

References

  1. Anđelic, M., & da Fonseca, C. M. (2021). On the constant coefficients of a certain recurrence relation: A simple proof. Heliyon, 7(8), Article ID e07764.
  2. Anđelic, M., da Fonseca, C. M., & Yılmaz, F. (2022). The bi-periodic Horadam sequence and some perturbed tridiagonal 2-Toeplitz matrices: A unified approach. Heliyon, 8(2), Article ID e08863.
  3. Bilgici, G. (2014). Two generalizations of Lucas sequence. Applied Mathematics and Computation, 245, 526–538.
  4. Burcu Bozkurt, Ş., Yılmaz, F., & Bozkurt, D. (2011). On the complex factorization of the Lucas sequence. Applied Mathematics Letters, 24, 1317–1321.
  5. Cahill, N. D., Derrico, J. R., & Spence, J. (2003). Complex factorizations of the Fibonacci and Lucas numbers. The Fibonacci Quarterly, 41(1), 13–19.
  6. Cahill, N. D., & Narayan, D. A. (2004). Fibonacci and Lucas numbers as tridiagonal matrix determinants. The Fibonacci Quarterly, 42(3), 216–221.
  7. Cooper, C., & Parry, R. Jr. (2004). Factorizations of some periodic linear recurrence systems. In: Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications, Germany, July 2004.
  8. da Fonseca, C. M., Kızılates¸, C., & Terzioglu, N. (2023). A second-order difference equation with sign-alternating coefficients. Kuwait Journal of Science, 50(2A): 1–8.
  9. da Fonseca, C. M., & Kowalenko, V. (2020). Eigenpairs of a family of tridiagonal matrices: Three decades later. Acta Mathematica Hungarica, 160, 376–389.
  10. da Fonseca, C. M., & Petronilho, J. (2001). Explicit inverses of some tridiagonal matrices. Linear Algebra and its Applications, 325(1–3), 7–21.
  11. da Fonseca, C. M., & Petronilho, J. (2005). Explicit inverse of a tridiagonal k-Toeplitz matrix. Numerische Mathematik, 100(3), 457–482.
  12. Edson, M., & Yayenie, O. (2009). A new generalization of Fibonacci sequences and extended Binet’s formula. Integers, 9(A48), 639–654.
  13. Feng, J. (2011). Fibonacci identities via the determinant of tridiagonal matrix. Applied Mathematics and Computation, 217, 5978–5981.
  14. Jun, S. P. (2015). Complex factorizations of the generalized Fibonacci sequences qn. Korean Journal of Mathematics, 23(3), 371–377.
  15. Nalli, A., & Civciv, H. (2009). A generalization of tridiagonal matrix determinants, Fibonacci and Lucas numbers. Chaos, Solitons & Fractals, 40, 355–361.
  16. Şahin, M., Tan, E., & Yılmaz, S. (2018). Complex factorization by Chebysev polynomials. Matematiche, 73, 179–189.
  17. Strang, G., & Borre, K. (1997). Linear Algebra, Geodesy and GPS. Wellesley, MA: Wellesley-Cambridge, pp. 555–577.
  18. Wu, H. (2014). Complex factorizations of the Lucas sequences via matrix methods. Journal of Applied Mathematics, 2014, Article ID 387675.
  19. Yılmaz, N., Coşkun, A., & Taskara, N. (2017). On properties of bi-periodic Fibonacci and Lucas polynomials. AIP Conference Proceedings, 1863, Article ID 310002.

Manuscript history

  • Received: 19 October 2024
  • Revised: 7 January 2025
  • Accepted: 8 May 2025
  • Online First: 9 May 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).

Related papers

Cite this paper

Shi, B., & Kızılateş, C. (2025). An algorithm for complex factorization of the bi-periodic Fibonacci and Lucas polynomials. Notes on Number Theory and Discrete Mathematics, 31(2), 269-279, DOI: 10.7546/nntdm.2025.31.2.269-279.

Comments are closed.