Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions

TL;DR

This paper establishes general criteria using differential Galois theory to determine when solutions of second-order difference equations are differentially transcendental, with applications to elliptic hypergeometric functions.

Contribution

It introduces new, uniform criteria for differential transcendence of solutions to various second-order difference equations, including elliptic hypergeometric functions.

Findings

Most elliptic hypergeometric functions are differentially transcendental.

Criteria apply to shift, q-dilation, Mahler, and elliptic difference equations.

Differential Galois theory underpins the developed criteria.

Abstract

We develop general criteria that ensure that any non-zero solution of a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases of interest, such as shift difference equations, q-dilation difference equations, Mahler difference equations, and elliptic difference equations. These criteria are obtained as an application of differential Galois theory for difference equations. We apply our criteria to prove a new result to the effect that most elliptic hypergeometric functions are differentially transcendental.