Differential Galois Theory of Linear Difference Equations

TL;DR

This paper develops a Galois theory for linear difference equations to analyze differential dependencies among solutions, reprove classical results, and establish new non-existence results for differential relations among special functions.

Contribution

It introduces a differential Galois theory for difference equations and applies it to prove the non-existence of certain differential relations among solutions.

Findings

Reproves Hölder's theorem on the Gamma function

Shows no differential relations among solutions of certain q-hypergeometric functions

Establishes a framework for analyzing differential dependencies in difference equations

Abstract

We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hoelder's Theorem that the Gamma function satisfies no polynomial differential equation and are able to give general results that imply, for example, that no differential relationship holds among solutions of certain classes of q-hypergeometric functions.