Sums of Powers of Fibonacci and Lucas Polynomials in terms of Fibopolynomials

TL;DR

This paper investigates sums of powers of Fibonacci and Lucas polynomials, providing conditions under which these sums can be expressed as linear combinations of specific Fibopolynomials, enhancing understanding of their algebraic structure.

Contribution

It introduces new conditions for representing sums of Fibonacci and Lucas polynomial powers as linear combinations of s-Fibopolynomials.

Findings

Derived sufficient conditions for sum representations

Expressed sums as linear combinations of s-Fibopolynomials

Analyzed both regular and alternating sums

Abstract

We study sums of powers of Fibonacci and Lucas polynomials of the form $% \sum_{n=0}^{q}F_{tsn}^{k}(x) $ and $\sum_{n=0}^{q}L_{tsn}^{k}% (x) $, where $s,t,k$ are given natural numbers, together with the corresponding alternating sums $\sum_{n=0}^{q}(-1) ^{n}F_{tsn}^{k}(x) $ and $\sum_{n=0}^{q}(-1) ^{n}L_{tsn}^{k}(x) $. We give sufficient conditions on the parameters $s,t,k$ for express these sums as linear combinations of certain $s$-Fibopolynomials.