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. Then we provide explicit formulas and determinantal expressions. Finally, we apply these results to recurrent polynomials of order $2$, we present several relations and interesting identities involving the Fibonacci polynomials of order $2$, the Lucas polynomials of order $2$, the classical Fibonacci polynomials, the classical Lucas polynomials, the Fibonacci numbers, the Lucas numbers, the Dickson polynomials of the first kind, and the Dickson polynomials of the second kind. Our results are a unified generalization of several works. Some well known results are special cases of ours.