New hypergeometric connection formulae between Fibonacci and Chebyshev polynomials

TL;DR

This paper introduces new hypergeometric-based connection formulae linking Fibonacci and Chebyshev polynomials, leading to novel expressions for Fibonacci numbers and related integrals.

Contribution

It presents novel connection formulae between Fibonacci and Chebyshev polynomials using hypergeometric functions, expanding the analytical tools for these classical sequences.

Findings

New hypergeometric connection formulae established

Derived new expressions for Fibonacci numbers

Evaluated integrals involving Fibonacci and Chebyshev polynomials

Abstract

We establish new connection formulae between Fibonacci polynomials and Chebyshev polynomials of the first and second kinds. These formulae are expressed in terms of certain values of hypergeometric functions of the type 2F1. Consequently, we obtain some new expressions for the celebrated Fibonacci numbers and their derivatives sequences. Moreover, we evaluate some definite integrals involving products of Fibonacci and Chebyshev polynomials.