An interesting generalized Fibonacci sequence: a two-by-two matrix representation

Abstract

The Fibonacci sequence is a well-known example of second order recurrence sequence, which belongs to a particular class of recursive sequences. In this article, other generalized Fibonacci sequence is introduced and defined by

$$\begin{aligned} F_{n+2}(\mu )=2aF_{n+1}(\mu )+(b-a^{2})F_{n}(\mu ),\ \ n\ge 0, \end{aligned}$$

where $F_{0}(\mu )=-\mu$, $F_{1}(\mu )=a^{2}-b-2a\mu$ and $\mu$ is a real number. Also n-th power of the generating matrix for this generalized Fibonacci sequence is established and some basic properties of this sequence are obtained by matrix methods.