# On certain rational recursive sequences of order four

## TL;DR

This paper derives solutions for a class of rational recursive sequences of order four using a group theoretic approach to reduce the order and solve the resulting lower order relations.

## Contribution

It introduces a novel group theoretic method to analyze and solve complex fourth-order recursive sequences with arbitrary coefficient sequences.

## Key findings

- Explicit solutions for the recursive sequences are obtained.
- The method simplifies the analysis of high-order rational recurrences.
- The approach can be applied to other complex recursive relations.

## Abstract

We obtain solutions to the recursive sequences of the form $$x_{n + 1} = \frac{x_{n - 3}x_{n }}{x_{n - 2}(a_n + b_nx_{n -3}x_{n})}$$ where $a_n$ and $b_n$ are arbitrary sequences of real numbers, and the initial values are gives as; $x_{-3},x_{-2}, x_{-1}$ and $x_{0}$. Our methodology is to employ a group theoretic method which lowers the order of the equations, and then solve the resulting lower order recurrence relations that arise therefrom.