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

In this study, a matrix R-v is defined, and two closed form expressions of the matrix R-v(n), for an integer n >= 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 Delta I)(n) are studied in terms of the generalized Fibonacci and Lucas numbers and binomial coefficients. By modifying results of 2 x 2 matrix representations given in the references of our study, we give various 3 x 3 matrix representations of the generalized Fibonacci and Lucas sequences. Many combinatorial identities are derived as applications.