Work

A Lindemann–Weierstrass theorem for 𝐸-functions

Year: 2024

Type: article

Abstract: 𝐸-functions were introduced by Siegel in 1929 to generalize Diophantine properties of the exponential function. After developments of Siegel’s methods by Shidlovskii, Nesterenko and André, Beukers proved in 2006 an optimal result on the algebraic independence of the values of 𝐸-functions which generalizes the Lindemann–Weierstrass theorem. Since then, it seems that no general result was stated concerning the relations between the values of a single 𝐸-function. We prove that André’s theory of 𝐸-operators and Beuker’s result lead to a Lindemann–Weierstrass theorem for 𝐸-functions in its “linear independence” formulation. As a consequence, we show that all transcendental values at algebraic arguments of an entire hypergeometric function are linearly independent over Q ̄ \overline{\mathbb{Q}} .

Source: Journal für die reine und angewandte Mathematik (Crelles Journal)

DOI: 10.1515/crelle-2024-0090

Author: Éric Delaygue

Institutions: Lyon 1 Université, Centre National de la Recherche Scientifique, Institut Camille Jordan

Language: English

Topic: Advanced Mathematical Identities

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Agence Nationale de la Recherche

Awards: G3010527862, Deciding irrationality and transcendence