A common generalization of Dickson polynomials, Fibonacci polynomials, and Lucas polynomials and applications

Said Zriaa\(^{a,∗}\), Mohammed Moucouf

\(^a\)Department of Mathematics, University Chouaib Doukkali, Eljadida Morocco

Correspondence should be addressed to: saidzriaa1992@gmail.com, moucouf@hotmail.com

Copyright © 2024 Said Zriaa, Mohammed Moucouf. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Published: Received: 23 January 2024 Accepted: 24 September 2024; Published Online: 26 September 2024

Abstract

In this work, we define a more general family of polynomials in several variables satisfying a linear recurrence relation. We provide explicit formulas and determinantal expressions. Our results are then applied to secondorder recurrent polynomials, presenting several relationships and identities involving Fibonacci polynomials of order 2, Lucas polynomials of order 2, classical Fibonacci polynomials, classical Lucas polynomials, Fibonacci numbers, Lucas numbers, and both kinds of Dickson polynomials. Our findings offer a unified generalization of various existing works, with several well-known results emerging as special cases.

Keywords:

Dickson polynomials, Fibonacci numbers, Lucas numbers, Fibonacci polynomials, Lucas polynomials.