# On certain sixth order difference equations

## TL;DR

This paper applies Lie group analysis to a sixth order difference equation, revealing its invariance properties and providing a unified approach to its solutions, including special cases with specific coefficient sequences.

## Contribution

It introduces a Lie group analysis method for solving a particular sixth order difference equation, highlighting its invariance properties and solution structure.

## Key findings

- The equation admits a two-dimensional Lie algebra.
- Solutions can be expressed in a unified form.
- Special cases with specific sequences are analyzed.

## Abstract

We use the Lie group analysis method to investigate the invariance properties and the solutions of \begin{align*} x_{n+1} =\frac{x_{n-5}x_{n-3}}{x_{n-1}(a_n +b_nx_{n-5}x_{n-3})}. \end{align*} We show that this equation has a two-dimensional Lie algebra and that its solutions can be presented in a unified manner. Besides presenting solutions of the recursive sequence above where $(a_n)$ and $(b_n)$ are sequences of real numbers, some specific cases are emphasized.