Extension of the Equation ${\sum }_{j=1}^{k}j{F}_j^p=F_n^q$ to a Family of Lucas Sequences

We solve the equation ${\sum }_{j=1}^{k}j{U}_j(x,y{)}^p=U_n(x,y{)}^q$ positive integers $x,p,q,k,n$, with $y=\pm 1$ and $\max \{p,q\}\le 11$, where $U_m(x,y)=\frac{{\alpha }^m-{\beta }^m}{\alpha -\beta }$ for $\alpha$ and $\beta$ roots of the polynomial $t^2-xt+y$. This generalizes existing results on similar equations, wherein the sequence was fixed as either the Fibonacci or Pell numbers. In addition, we find all solutions with $k=2$ and $y=\pm 1$.