Form of the solutions of difference equations via lie symmetry analysis and Fibonacci numbers

Abstract

In this paper, we give the form of the solutions of the following rational difference equation

^Xn + 1 = ^Xn - 3^Xn -1 / ^an^Xn - 3 + bn^Xn - 1

where a_n and b_n are sequences of real numbers, by using Lie symmetry analysis and associated with Fibonacci numbers, respectively. We find an interesting relation between the exact solution of Equation (1) and the classical Fibonacci sequence.