Extension of the Equation $\sum\limits_{j=1}^{k}jF_{j}^{p}=F_{n}^{q}$ to a Family of Lucas Sequences

Abstract

We solve the equation $\sum\limits_{j=1}^{k}jU_{j}(x,y)^{p}=U_{n}(x,y)^{q}$ positive integers $x,p,q,k,n$, with $y=\pm1$ and $\max\{p,q\}\leq11$, where $U_{m}(x,y)=\frac{α^{m}-β^{m}}{α-β}$ for $α$ and $β$ 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 $y=\pm1$.