Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions
Number Theory
2021-01-22 v3 Mathematical Physics
Commutative Algebra
math.MP
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.
Keywords
Cite
@article{arxiv.1809.05416,
title = {Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions},
author = {Carlos E. Arreche and Thomas Dreyfus and Julien Roques},
journal= {arXiv preprint arXiv:1809.05416},
year = {2021}
}