Some identities for the generalized Fibonacci polynomials by the Q(x) matrix

TL;DR

This paper derives new identities for generalized Fibonacci polynomials using the Q(x) matrix, including Cassini and Honsberger formulas, applicable to various polynomial sequences like Fibonacci, Lucas, and Pell polynomials.

Contribution

It introduces identities for generalized Fibonacci polynomials via the Q(x) matrix, extending known formulas to a broader class of polynomial sequences.

Findings

Derived identities including Cassini and Honsberger formulas

Applicable to multiple polynomial sequences such as Fibonacci, Lucas, and Pell

Provides a unified matrix-based approach for polynomial identities

Abstract

In this note, we obtain some identities for the generalized Fibonacci polynomial by using the Q(x) matrix. These identities including the Cassini identity and Honsberger formula can be applied to some polynomial sequences, such as Fibonacci polynomials, Lucas polynomials, Pell polynomials, Pell-Lucas polynomials, Fermat polynomials, Fermat-Lucas polynomials, and so on.