On invariants and forbidden sets

TL;DR

This paper introduces six new algebraic invariants for rational difference equations, enabling order reduction and the explicit determination of forbidden sets and solutions for complex initial conditions.

Contribution

The paper presents novel algebraic invariants that facilitate order reduction and explicit solutions for six classes of rational difference equations, including higher-order linear fractional cases.

Findings

Identified forbidden sets for all six cases.

Derived closed-form solutions for initial conditions outside forbidden sets.

Extended analysis to complex initial conditions and parameters.

Abstract

We introduce six new algebraic invariants for rational difference equations. We use these invariants to perform a reduction of order in each case. This reduction of order allows us to find forbidden sets in each case. These six cases include two linear fractional rational difference equations of order greater than one. In all six cases, we give a closed form solution for all initial conditions which are not in the forbidden set. In all six cases, the initial conditions and parameters are assumed to be arbitrary complex numbers.