A Method to Construct Generalized Fibonacci Sequences

Abstract

The main purpose of this paper is to study the convergence properties of Generalized Fibonacci Sequences and the series of partial sums associated with them. When the proper values of an $s\times s$ real matrix $A$ are real and different, we give a necessary and sufficient condition for the convergence of the matrix sequence $A,A^{\mathrm{2}},A^{\mathrm{3}},\dots$ to a matrix $B$.