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

TL;DR

This paper introduces a unified family of multivariable polynomials satisfying linear recurrences, generalizing classical Fibonacci, Lucas, and Dickson polynomials, with explicit formulas, identities, and applications to various polynomial families.

Contribution

It provides a comprehensive generalization of multiple polynomial families, offering explicit formulas, identities, and applications that unify and extend existing results.

Findings

Derived explicit formulas and determinantal expressions for the generalized polynomials.

Established relations and identities linking Fibonacci, Lucas, and Dickson polynomials.

Unified framework encompasses several classical polynomial families as special cases.

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.