Products of 2X2 matrices related to non autonomous Fibonacci difference equations

TL;DR

This paper develops a method to compute products of specific 2x2 Fibonacci matrices, enabling explicit solutions for non-autonomous Fibonacci difference equations, including periodic cases with Floquet analysis.

Contribution

It introduces a technique for arbitrary matrix products and derives explicit solutions for non-autonomous Fibonacci difference equations, including periodic cases.

Findings

Explicit solutions for non-autonomous Fibonacci difference equations.

Analysis of periodic cases using monodromy matrix and Floquet multipliers.

Development of a method to compute products of Fibonacci matrices.

Abstract

A technique to compute arbitrary products of a class of Fibonacci $2\times2$ square matrices is proved in this work. General explicit solutions for non autonomous Fibonacci difference equations are obtained from these products. In the periodic non autonomous Fibonacci difference equations the monodromy matrix, the Floquet multipliers and the Binet's formulas are obtained. In the periodic case explicit solutions are obtained and the solutions are analyzed.