# The invariance and formulas for solutions of some fifth-order difference equations

## TL;DR

This paper applies Lie group analysis to a class of fifth-order difference equations, deriving their symmetries and explicit solutions, thereby extending previous work on rational recursive sequences.

## Contribution

It introduces a symmetry-based method to find exact solutions for complex fifth-order difference equations, generalizing prior results.

## Key findings

- Derived non-trivial symmetries of the difference equations
- Obtained explicit formulas for solutions
- Extended previous results on rational recursive sequences

## Abstract

Lie group analysis of the difference equations of the form \begin{align*} x_{n+1} =\frac{x_{n-4}x_{n-3}}{x_{n}(a_n +b_nx_{n-4}x_{n-3}x_{n-2}x_{n-1})}, \end{align*} where $a_n$ and $b_n$ are real sequences, is performed and non-trivial symmetries are derived. Furthermore, we find find formulas for exact solutions of the equations. This work generalizes a recent result by Elsayed [Elsayed, E.M.: {Expression and behavior of the solutions of some rational recursive sequences}. {Math. Meth. Appl. Sci.} { \bf 2016:39},5682--5694 (2016)].