Novel Results of Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals

TL;DR

This paper introduces new connection formulae between generalized Fibonacci and Lucas polynomials involving hypergeometric functions, leading to novel formulas for special numbers and radical reductions.

Contribution

It develops new connection formulae between generalized Fibonacci and Lucas polynomials using hypergeometric functions, extending existing literature and enabling radical reduction formulas.

Findings

New connection formulae involving hypergeometric functions

Generalized formulas linking Fibonacci, Lucas, Pell, Fermat polynomials

Radical reduction formulas for odd and even radicals

Abstract

This paper is concerned with developing some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. All the connection coefficients involve hypergeometric functions of the type $_2F_{1}(z)$, for certain $z$. Several new connection formulae between some famous polynomials such as Fibonacci, Lucas, Pell, Fermat, Pell-Lucas, and Fermat-Lucas polynomials are deduced as special cases of the derived connection formulae. Some of the introduced formulae generalize some of those existing in the literature. As two applications of the derived connection formulae, some new formulae linking some celebrated numbers are given and also some newly closed formulae of certain definite weighted integrals are deduced. Based on using the two generalized classes of Fibonacci and Lucas polynomials, some new reduction formulae of certain odd and even radicals are developed.