A Lindemann-Weierstrass theorem for $E$-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 $\overline{\mathbb{Q}}$.