# On a system of difference equations of second order solved in a closed from

## TL;DR

This paper derives closed-form solutions for a specific second-order difference system, explores particular cases involving Tribonacci and Padovan numbers, and establishes the global stability of positive equilibrium points, extending recent research findings.

## Contribution

It provides explicit solutions for a complex difference system and analyzes stability, including special cases with well-known number sequences, which was not previously addressed.

## Key findings

- Closed-form solutions for the difference system.
- Representation of solutions using Tribonacci and Padovan numbers.
- Proof of global stability of positive equilibrium points.

## Abstract

In this work we solve in closed form the system of difference equations \begin{equation*} x_{n+1}=\dfrac{ay_nx_{n-1}+bx_{n-1}+c}{y_nx_{n-1}},\; y_{n+1}=\dfrac{ax_ny_{n-1}+by_{n-1}+c}{x_ny_{n-1}},\;n=0,1,..., \end{equation*} where the initial values $x_{-1}$, $x_0$, $y_{-1}$ and $y_0$ are arbitrary nonzero real numbers and the parameters $a$, $b$ and $c$ are arbitrary real numbers with $c\ne 0$. In particular we represent the solutions of some particular cases of this system in terms of Tribonacci and Padovan numbers and we prove the global stability of the corresponding positive equilibrium points. The result obtained here extend those obtained in some recent papers.