The Representations of the Fibonacci and Lucas Matrices

Abstract

In this study, a matrix $R_{L}$ is defined by the properties associated with the Pascal matrix, and two closed-form expressions of the matrix function $f(R_{L})=R_{L}^{n}$ are determined by methods in matrix theory. These expressions satisfy a connection between the integer sequences of the first–second kinds and the Pascal matrices. The matrix $R_{L}^{n}$ is the Fibonacci Lucas matrix, whose entries are the Fibonacci and Lucas numbers. Also, the representations of the Lucas matrix are derived by the matrix function $f(R_{L}-5I)$, and various forms of the matrix $(R_{L}-5I)^{n}$ in terms of a binomial coefficient are studied by methods in number theory. These representations give varied ways to obtain the new Fibonacci- and Lucas-type identities via several properties of the matrices $R_{L}^{n}$ and $(R_{L}-5I)^{n}$.