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

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.