On the Powers of the $k$-Fibonacci Numbers

Abstract

In this paper, we will find a combinatorial formula that relates the power of a $k$-Fibonacci number, $F_{k,n}^p$, to the number $F_{k,an}$. From this formula, and if $p$ is odd, we will find a new formula that allows expressing the $k$-Fibonacci number $F_{k,(2r+1)n}$ as a combination of odd powers of $F_{k,n}$. If $p$ is even, the formula is similar but for the even $k$-Lucas numbers $L_{k,2rn}$.