Arithmetic theory of E-operators

TL;DR

This paper explores the arithmetic properties of E-operators related to E-functions, analyzing connection and Stokes constants, and introduces a class of numbers with algebraic approximations motivated by fundamental constants.

Contribution

It provides a detailed study of the arithmetical nature of connection and Stokes constants of E-operators, linking them to values of E-functions, G-functions, and derivatives of the Gamma function.

Findings

Connection constants involve values of E-functions at algebraic points.

Stokes constants relate to values of G-functions and derivatives of Gamma at rational points.

Introduces a class of numbers with algebraic approximations inspired by fundamental constants.

Abstract

In [S\'eries Gevrey de type arithm\'etique I Th\'eor\'emes de puret\'e et de dualit\'e, Annals of Math. 151 (2000), 705--740], Andr\'e has introduced E-operators, a class of differential operators intimately related to E-functions, and constructed local bases of solutions for these operators. In this paper we investigate the arithmetical nature of connexion constants of E-operators at finite distance, and of Stokes constants at infinity. We prove that they involve values at algebraic points of E-functions in the former case, and in the latter one, values of G-functions and of derivatives of the Gamma function at rational points in a very precise way. As an application, we define and study a class of numbers having certain algebraic approximations defined in terms of E-functions. These types of approximations are motivated by the convergents to the number e, as well as by recent constructions of approximations to Euler's constant and values of the Gamma function. Our results and methods are completely different from those in our paper [On the values of G-functions, Commentarii Math. Helv., to appear], where we have studied similar questions for G-functions.