Chebyshev-Fibonacci polynomial relations using generating functions

TL;DR

This paper explores the mathematical relationships between Chebyshev polynomials and Fibonacci polynomials through generating functions, leading to new combinatorial identities involving balancing polynomials and Fibonacci numbers.

Contribution

It introduces novel connections between Chebyshev and Fibonacci polynomials via generating functions, expanding the understanding of their interrelations.

Findings

Established new identities linking Chebyshev and Fibonacci polynomials.

Derived relations between their generating functions.

Presented combinatorial identities involving balancing polynomials and Fibonacci numbers.

Abstract

The main object of the paper is to reveal connections between Chebyshev polynomials of the first and second kinds and Fibonacci polynomials introduced by Catalan. This is achieved by relating the respective (ordinary and exponential) generating functions to each other. As a consequence, we also establish new combinatorial identities for balancing polynomials and Fibonacci (Lucas) numbers.