On The Dynamics Of Solutions Of A Rational Difference Equation Via Generalized Tribonacci Numbers

TL;DR

This paper analyzes the solutions, stability, and long-term behavior of a specific rational difference equation linked to generalized Tribonacci numbers, providing insights into its mathematical properties and solution dynamics.

Contribution

It introduces a detailed study of a rational difference equation's solutions, stability, and asymptotics, connecting them with generalized Tribonacci numbers for the first time.

Findings

Solutions are explicitly characterized in terms of generalized Tribonacci numbers.

The stability conditions of the solutions are established.

Asymptotic behavior of solutions is described under various parameter settings.

Abstract

In this study, we investigate the form of solutions, stability character and asymptotic behavior of the following rational difference equation x_{n+1}=({\gamma}/(x_{n}(x_{n-1}+{\alpha})+\b{eta})), n=0,1,..., where the inital values x_{-1} and x_{0} and {\alpha}, \b{eta} and {\gamma} with {\gamma} are nonnegative real numbers. Its solutions are associated with generalized Tribonacci numbers.