On the sum of powers of two $k$-Fibonacci numbers which belongs to the sequence of $k$-Lucas numbers

Abstract

Let $k\geq 1$ and denote $(F_{k,n})_{n\geq 0}$, the $k$-Fibonacci sequence whose terms satisfy the recurrence relation $F_{k,n}=kF_{k,n-1}+F_{k,n-2}$,

with initial conditions $F_{k,0}=0$ and $F_{k,1}=1$. In the same way, the $k$-Lucas sequence $(L_{k,n})_{n\geq 0}$ is defined by satisfying the same recurrence relation with initial values $L_{k,0}=2$ and $L_{k,1}=k$. These sequences was introduced by Falcon and Plaza and they showed many of its properties too. In particular, they proved that $F_{k,n+1}+F_{k,n-1}=L_{k,n}$, for all $k\geq 1$ and $n\geq 0$. In this paper, we shall prove that if $k\geq1$ and $F_{k,n+1}^s+F_{k,n-1}^s\in (L_{k,m})_{m\geq 1}$ for infinitely many positive integers $n$, then $s=1$.