On Rational and Hypergeometric Solutions of Linear Ordinary Difference Equations in $\Pi\mathbf\Sigma^*$-field extensions

TL;DR

This paper introduces a comprehensive algorithm for finding all hypergeometric and rational solutions of linear difference equations within a broad class of difference fields, enabling solutions in terms of nested sums and products.

Contribution

It provides a complete, flexible framework for solving linear difference equations in $ ext{Pi} ext{Sigma}^*$-field extensions, extending previous methods to a wider class of difference fields.

Findings

Algorithm computes all hypergeometric solutions.

Algorithm computes all rational solutions.

Framework applies to a broad class of difference fields.

Abstract

We present a complete algorithm that computes all hypergeometric solutions of homogeneous linear difference equations and rational solutions of parameterized linear difference equations in the setting of $\Pi\Sigma^*$-fields. More generally, we provide a flexible framework for a big class of difference fields that is built by a tower of $\Pi\Sigma^*$-field extensions over a difference field that satisfies certain algorithmic properties. As a consequence one can compute all solutions in terms of indefinite nested sums and products that arise within the components of a parameterized linear difference equation, and one can find all hypergeometric solutions that are defined over the arising sums and products of a homogeneous linear difference equation.