Irreducibility of generalized Fibonacci polynomials

Abstract

A second order polynomial sequence is of Fibonacci-type ${\mathcal{F}}_n$ (Lucas-type ${\mathcal{L}}_n$) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. Under certain conditions these polynomials are irreducible if and only if $n$ is a prime number. For example, the Fibonacci polynomials, Pell polynomials, Fermat polynomials, Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials are irreducible when $n$ is a prime number; and Chebyshev polynomials (second kind), Morgan-Voyce polynomials (Fibonacci type), and Vieta polynomials are reducible when $n$ is a prime number. In this paper we give some theorems to determine whether the Fibonacci type polynomials and Lucas type polynomials are irreducible when $n$ is prime.

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