Diophantine equation with weighted k-Fibonacci numbers

Assume that p, q, and k are integers for which the conditions \(1\le p,q\le 10\) $1\le p,q\le 10$ and \(2\le k\le 10\) $2\le k\le 10$ are satisfied. The initial values \(G_0=0\) $G_0=0$ , \(G_1=1\) $G_1=1$ , together with the recursive rule \(G_m=kG_{m-1}+G_{m-2}\) $G_m=k{G}_{m-1}+G_{m-2}$ define the non-negative integer sequence \(\{G_m\}_{m=0}^\infty\) ${\{G_m\}}_{m=0}^{\infty }$ . In this paper, we solve completely the diophantine equation \(\begin{aligned} G_1^p+2G_2^p+\cdots +\ell G_\ell ^p=G_n^q \end{aligned}\) $$\begin{array}{c}\hfill G_1^p+2G_2^p+\cdots +\ell G_{\ell }^p=G_n^q\end{array}$$ in the positive integers \(k,p,q,\ell ,n\) $k,p,q,\ell ,n$ unconditionally for \(\ell\) $\ell$ and n. The method works, at least in theory for arbitrary positive integers p, q, and k.