The Power Sums Involving Fibonacci Polynomials and Their Applications

Li Chen,Xiao Wang

doi:10.3390/sym11050635

Abstract

The Girard and Waring formula and mathematical induction are used to study a problem involving the sums of powers of Fibonacci polynomials in this paper, and we give it interesting divisible properties. As an application of our result, we also prove a generalized conclusion proposed by R. S. Melham.

Highlights

For any integer n ≥ 0, the famous Fibonacci polynomials { Fn ( x )} and Lucas polynomials{ Ln ( x )} are defined as F0 ( x ) = 0, F1 ( x ) = 1, L0 ( x ) = 2, L1 ( x ) = x√and Fn+2 ( x ) = xF√n+1 ( x ) + Fn ( x ), Ln+2 ( x ) = xLn+1 ( x ) + Ln ( x ) for all n ≥ 0
Since the Fibonacci numbers and Lucas numbers occupy significant positions in combinatorial mathematics and elementary number theory, they are studied by plenty of researchers, and have gained a large number of vital conclusions; some of them can be found in References [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]
Yi Yuan and Zhang Wenpeng [1] studied the properties of the Fibonacci polynomials, and proved some interesting identities involving Fibonacci numbers and Lucas numbers

Summary

Introduction

For any integer n ≥ 0, the famous Fibonacci polynomials { Fn ( x )} and Lucas polynomials. Yi Yuan and Zhang Wenpeng [1] studied the properties of the Fibonacci polynomials, and proved some interesting identities involving Fibonacci numbers and Lucas numbers. Kiyota Ozeki [3] got some identity involving sums of powers of Fibonacci numbers Sun et al in [7] proved that this polynomial and its derivative both disappear at 1, and it can be an integer polynomial when a product of the first consecutive Lucas numbers of odd order multiplies it. This presents an affirmative answer to Conjecture 1 of Melham. It is clear that our Corollary 1 gave a new proof for Conjecture 1

Several Lemmas

Proof of the Theorem