A Lindemann-Weierstrass theorem for $E$-functions

TL;DR

This paper extends the Lindemann-Weierstrass theorem to $E$-functions, demonstrating their linear independence at algebraic points and revealing new transcendence properties of hypergeometric functions.

Contribution

It establishes a Lindemann-Weierstrass type theorem for $E$-functions using André's $E$-operator theory and Beuker's algebraic independence results.

Findings

Proves linear independence of $E$-function values at algebraic points.

Shows transcendental values of hypergeometric functions are linearly independent.

Extends classical transcendence results to a broader class of functions.

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 Andr\'e, 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 Andr\'e'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 $\overline{\mathbb{Q}}$.