A three by three Pascal matrix representations of the generalized Fibonacci and Lucas sequences

Abstract

In this study, a matrix $R_{v}$ is defined, and two closed form expressions of the matrix $R_{v}^{n}$, for an integer $n\geq 1$, are evaluated by the matrix functions in matrix theory. These expressions satisfy a connection between the generalized Fibonacci and Lucas numbers with the Pascal matrices. Thus, two representations of the matrix $R_{v}^{n}$ and various forms of matrix $(R_{v}+q\triangle I)^{n}$ are studied in terms of the generalized Fibonacci and Lucas numbers and binomial coefficients. By modifying results of $2\times 2$ matrix representations given in the references of our study, we give various $3\times 3$ matrix representations of the generalized Fibonacci and Lucas sequences. Many combinatorial identities are derived asapplications.