On Generalized Fibonacci Polynomials: Horadam Polynomials

  • Yüksel Soykan Department of Mathematics, Science Faculty, Zonguldak Bülent Ecevit University, 67100, Zonguldak, Turkey
Keywords: Horadam polynomials, (r, s)-Fibonacci polynomials, (r, s)-Lucas polynomials, generalized Fibonacci polynomials, Fibonacci polynomials, Lucas polynomials

Abstract

In this paper, we investigate the generalized Fibonacci (Horadam) polynomials and we deal with, in detail, two special cases which we call them $(r,s)$-Fibonacci and $(r,s)$-Lucas polynomials. We present Binet's formulas, generating functions, Simson's formulas, and the summation formulas for these polynomial sequences. Moreover, we give some identities and matrices associated with these sequences. Finally, we present several expressions and combinatorial results of the generalized Fibonacci polynomials.

References

D. Andrica and O. Bagdasar, Recurrent Sequences Key Results, Applications, and Problems, Springer, 2020. https://doi.org/10.1007/978-3-030-51502-7

G.B. Djordjević and G.V. Milovanović, Special classes of polynomials, University of Niš, Faculty of Technology, Leskovac, 2014. http://www.mi.sanu.ac.rs/~gvm/Teze/Special%20Classes%20of%20Polynomials.pdf

G. Frei, Binary Lucas and Fibonacci polynomials, I, Math. Nachr. 96 (1980), 83-112 https://doi.org/10.1002/mana.19800960109

R. Flórez, N. McAnally and A. Mukherjee, Identities for the generalized Fibonacci polynomial, Integers 18B (2018), Paper No. A2, 13 pp.

T.-X. He and Peter J.-S. Shiue, On sequences of numbers and polynomials defined by linear recurrence relations of order 2, International Journal of Mathematics and Mathematical Sciences 2009 (2009), Article ID 709386, 21 pp. https://doi.org/10.1155/2009/709386

T. Koshy, Pell and Pell-Lucas Numbers with Applications, Springer, New York, 2014. https://doi.org/10.1007/978-1-4614-8489-9

T. Koshy, Fibonacci and Lucas numbers with applications, Volume 1, Pure and Applied Mathematics : A Wiley Series of Texts, Monographs and Tracts, Second Edition, John Wiley & Sons, New York, 2018.

T. Koshy, Fibonacci and Lucas numbers with applications, Volume 2, Pure and Applied Mathematics : A Wiley Series of Texts, Monographs and Tracts, John Wiley & Sons, New York, 2019.

Y. Soykan, Simson identity of generalized m-step Fibonacci numbers, Int. J. Adv. Appl. Math. and Mech. 7(2) (2019), 45-56.

S. Vajda, Fibonacci and Lucas Numbers, and the Golden Section : Theory and Applications, Dover Publications Inc., 2008.

J. Wang, Some new results for the (p, q)-Fibonacci and Lucas polynomials, Advances in Difference Equations 2014 (2014), 64. https://doi.org/10.1186/1687-1847-2014-64

W. Wang and H. Wang, Generalized-Humbert polynomials via generalized Fibonacci polynomials, Applied Mathematics and Computation 307 (2017), 204-216. https://doi.org/10.1016/j.amc.2017.02.050

Published
2022-09-09
How to Cite
Soykan, Y. (2022). On Generalized Fibonacci Polynomials: Horadam Polynomials. Earthline Journal of Mathematical Sciences, 11(1), 23-114. https://doi.org/10.34198/ejms.11123.23114
Section
Articles