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

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.