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arXiv · 2212.00388

Hypertranscendence and linear difference equations, the exponential case

Thomas Dreyfus
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Abstract

In this paper we study meromorphic functions solutions of linear shift difference equations in coefficients in $\mathbb{C}(x)$ involving the operator $ρ: y(x)\mapsto y(x+h)$, 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.

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Thomas Dreyfus. 2025-09-05. Hypertranscendence and linear difference equations, the exponential case. https://doi.org/10.1016/j.jalgebra.2025.08.025

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