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 \(L_{k,2rn}\).