On some properties of k-generalized Fibonacci numbers

Abstract

Some identities between the positive integer powers of the matrices $X$ satisfying the equation $X^{k+1}-(r+1)X^k+rI+\sum\limits_{i=1}^{k-1}(r-1)X^i=0$ and the $k$-generalized Fibonacci numbers are first established, where $k$ is an integer with $k\geq 2$. Then, two main formulas, one of which is related to the $k$-generalized Fibonacci sequence, are given.