Hypertranscendence and linear difference equations

TL;DR

This paper establishes comprehensive results demonstrating that solutions to various linear difference equations, including shift, q-difference, and Mahler operators, are hypertranscendental, extending classical theorems and using parametrized difference Galois theory.

Contribution

It provides the first complete characterization of hypertranscendence for solutions to general linear difference equations associated with key operators, using advanced Galois theory methods.

Findings

Solutions to these difference equations are hypertranscendental.

Results apply to solutions expressed as Laurent series in the variable x.

Algebraic independence of Mahler function values and derivatives at algebraic points is established.

Abstract

After H\"older proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental i.e. they cannot be solution to an algebraic differential equation). In this paper, we obtain the first complete results for solutions to general linear difference equations associated with the shift operator $x\mapsto x+h$ ($h\in\mathbb{C}^*$), the $q$-difference operator $x\mapsto qx$ ($q\in\mathbb{C}^*$ not a root of unity), and the Mahler operator $x\mapsto x^p$ ($p\geq 2$ integer). The only restriction is that we constrain our solutions to be expressed as (possibly ramified) Laurent series in the variable $x$ with complex coefficients (or in the variable $1/x$ in some special case associated with the shift operator). Our proof is based on the parametrized difference Galois theory initiated by Hardouin and Singer. We also deduce from our main result a general statement about algebraic independence of values of Mahler functions and their derivatives at algebraic points.