On (k,p)-Fibonacci numbers and matrices

In this paper, some relations between the powers of any matrices $X$ satisfying the equation $X^k-pX^{k-1}-(p-1)X-I=\bf{0}$ and $(k,p)$-Fibonacci numbers are established with $k\geq2$. First, a result is obtained to find the powers of the matrices satisfying the condition above via $(k,p)$-Fibonacci numbers. Then, new properties related to $(k,p)$-Fibonacci numbers are given. Moreover, some relations between the sequence $\{F_{3,s}(n)\}$ and the generalized Fibonacci sequence $\{U_n(p,q)\}$ are also examined.