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 $$(R_{L}-5I)^{n}$$ .