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 and ${\sum }_{n=0}^q{L}_{tsn}^k$, 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.

arXiv comment16 pages. Revised and shortened version