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

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) $\mathrm{.}Wegivesufficientconditionsontheparameters$s,t,k$forexpressthesesumsaslinearcombinationsofcertain$s$-Fibopolynomials.