Le th\'eor\`eme d'Andr\'e-Chudnovsky-Katz {\guillemotleft} au sens large {\guillemotright}

TL;DR

This paper extends the Andre9-Chudnovsky-Katz theorem to a broader context for $E$- and $G$-functions, providing new structural insights and a novel proof of a generalized algebraic independence result.

Contribution

It establishes a broad sense version of the Andre9-Chudnovsky-Katz theorem and applies it to prove a generalized algebraic independence of $E$-function values.

Findings

Structural theorem for broad sense $G$-operators

Structural theorem for broad sense $E$-operators

New proof of generalized Siegel-Shidlovskii theorem

Abstract

Siegel's $E$- and $G$-functions were defined in two conjecturally equivalent senses, strict and broad. By taking up and completing a sketch of Andr\'e, we state and prove the analogue in the broad sense of the Andr\'e-Chudnovsky-Katz theorem, which is a structure theorem on the $G$-operators in the broad sense (they are differential operators cancelling the $G$-functions in the strict sense). We deduce from that a structure theorem on the $E$-operators in the broad sense, which are differential operators cancelling the $E$-functions in the broad sense. As an application of this last theorem, we give a new proof of a generalization by Andr\'e of the Siegel-Shidlovskii theorem on the algebraic independence of the values of $E$-functions in the broad sense.