Some notes on a Fibonacci-Lucas identity

TL;DR

This paper explores new polynomial identities involving generalized Lucas sequences, extending previous Fibonacci-Lucas identities and connecting them to Chebyshev polynomials, with emphasis on novel cases and special instances.

Contribution

It introduces additional polynomial identities for generalized Lucas sequences and establishes their analogues for Chebyshev polynomials, expanding the mathematical understanding of these sequences.

Findings

New polynomial identities for generalized Lucas sequences

Extensions of Fibonacci-Lucas identities to broader contexts

Polynomial analogues for Chebyshev polynomials

Abstract

In 2016, Edgar and, independently of him, Bhatnagar sta\-ted a nice polynomial identity that connects Fibonacci and Lucas numbers. Shortly after their publications, this identity has been generalized in two different ways: Dafnis, Phillipou and Livieris provided a generalization to Fibonacci sequences of order $k$ and Abd-Elhameed and Zeyada extended Edgar--Bhatnagar identity to generalized Fibonacci and Lucas sequences. In this paper, we present more polynomial identities for generalized Lucas sequences. We discuss interesting aspects and special cases which have not been stated before but deserve recognition. Finally, we prove the polynomial analogues of these identities for Chebyshev polynomials.