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.
Highlights
The differential Galois theory for difference equations developed in [HS08] provides a theoretical tool to understand the differential-algebraic properties of solutions of linear difference equations
The theory of [HS08] associates with (1.1) a geometric object G, called the differential Galois group, that encodes the polynomial differential equations satisfied by the solutions of (1.1)
In each of the four cases of interest mentioned above there are effective algorithms to decide whether the Riccati equation (1.3) and the telescoping problem (1.4) admit solutions, for which we provide case-by-case references below. These user-friendly criteria are of practical import to non-experts seeking to decide differential transcendence of solutions of second-order difference equations in many settings that arise in applications
Summary
The differential Galois theory for difference equations developed in [HS08] provides a theoretical tool to understand the differential-algebraic properties of solutions of linear difference equations. In each of the four cases of interest mentioned above there are effective algorithms to decide whether the Riccati equation (1.3) and the telescoping problem (1.4) admit solutions, for which we provide case-by-case references below These user-friendly criteria are of practical import to non-experts seeking to decide differential transcendence of solutions of second-order difference equations in many settings that arise in applications. The general criteria developed in [DHR21, DHR18] for differential transcendence of solutions of (1.1) are valid for arbitrary n, but these criteria require prior knowledge of the (non-differential) Galois group H of (1.1) [vdPS97] At present this group H can only be computed in general when n 2 by [Hen97] in the q-dilation case and [Roq18] in the Mahler case. The first author thanks Professor Spiridonov for helpful discussions on an earlier version of this work, which led to the important splitting of Theorems 5.6 and 5.7 into cases (A) and (B)
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