Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation

TL;DR

This paper develops algorithms to compute the difference-differential Galois group of second-order linear difference equations and uses it to determine the differential relations among solutions.

Contribution

It introduces a method to compute the Galois group for such equations and deduces solution relations from the group's defining equations.

Findings

Algorithms for Galois group computation

Determination of differential relations among solutions

Application to second-order linear difference equations

Abstract

We apply the difference-differential Galois theory developed by Hardouin and Singer to compute the differential-algebraic relations among the solutions to a second-order homogeneous linear difference equation of the form $ y(x+2)+a(x)y(x+1)+b(x)y(x)=0,$ where the coefficients $a(x),b(x)\in \bar{\mathbb{Q}}(x)$ are rational functions in $x$ with coefficients in $\bar{\mathbb{Q}}$. We develop algorithms to compute the difference-differential Galois group associated to such an equation, and show how to deduce the differential-algebraic relations among the solutions from the defining equations of the Galois group.