Arithmetic theory of q-difference equations (G_q-functions and q-difference modules of type G, global q-Gevrey series)

TL;DR

This paper develops an arithmetic framework for q-difference equations, defining G_q-functions, proving their regularity, and exploring the irrationality of certain special values through q-analogues of classical theorems and Fourier transformations.

Contribution

It introduces the concept of G_q-functions, establishes their regularity, and connects them with q-analogues of key theorems and Fourier analysis to study special value irrationality.

Findings

G_q-functions are regular under certain conditions.

Formal q-Borel transformations of G_q-functions have irrational special values.

The paper extends classical theorems to the q-difference setting.

Abstract

In the first part of the paper we give a definition of G_q-function and we establish a regularity result, obtained as a combination of a q-analogue of the Andre'-Chudnovsky Theorem [And89, VI] and Katz Theorem [Kat70, \S 13]. In the second part of the paper, we combine it with some formal q-analogous Fourier transformations, obtaining a statement on the irrationality of special values of the formal $q$-Borel transformation of a G_q-function.