Hypertranscendence and linear difference equations, the exponential case

TL;DR

This paper investigates solutions of linear shift difference equations with meromorphic functions, showing that solutions satisfying algebraic differential equations are composed of periodic functions and exponentials, using parametrized difference Galois theory.

Contribution

It establishes a link between solutions of difference equations and differential algebraic properties, applying parametrized Galois theory to characterize solutions.

Findings

Solutions of certain difference equations are contained in rings generated by exponentials and periodic functions.

If a solution satisfies an algebraic differential equation, it has a specific algebraic structure.

The proof utilizes advanced Galois theory for difference equations.

Abstract

In this paper we study meromorphic functions solutions of linear shift difference equations in coefficients in $\mathbb{C}(x)$ involving the operator $\rho: y(x)\mapsto y(x+h)$, for some $h\in \mathbb{C}^*$. We prove that if $f$ is solution of an algebraic differential equation, then $f$ belongs to a ring that is made with periodic functions and exponentials. Our proof is based on the parametrized difference Galois theory initiated by Hardouin and Singer.