Identities for the generalized Fibonacci polynomial

TL;DR

This paper generalizes classical identities from Fibonacci and Lucas numbers to their polynomial counterparts, establishing relationships among various polynomial sequences of Fibonacci type and Lucas type.

Contribution

It introduces generalized identities for Fibonacci and Lucas polynomials, extending known number identities to polynomial sequences with similar recurrence relations.

Findings

Derived identities linking Fibonacci and Lucas polynomials.

Extended relationships among Pell, Fermat, Chebyshev, Jacobsthal, and Morgan-Voyce polynomials.

Unified framework for polynomial identities of Fibonacci type and Lucas type.

Abstract

A second order polynomial sequence is of Fibonacci type (Lucas type) if its Binet formula is similar in structure to the Binet formula for the Fibonacci (Lucas) numbers. In this paper we generalize identities from Fibonacci numbers and Lucas numbers to Fibonacci type and Lucas type polynomials. A Fibonacci type polynomial is equivalent to a Lucas type polynomial if they both satisfy the same recurrence relations. Most of the identities provide relationships between two equivalent polynomials. In particular, each type of identities in this paper relate the following polynomial sequences: Fibonacci with Lucas, Pell with Pell-Lucas, Fermat with Fermat-Lucas, both types of Chebyshev polynomials, Jacobsthal with Jacobsthal-Lucas and both types of Morgan-Voyce.