Arithmetic theory of E-operators

Stephane Fischler (1) , Tanguy Rivoal (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 IF - Institut Fourier

In [Séries Gevrey de type arithmétique I Théorémes de pureté et de dualité, Annals of Math. 151 (2000), 705--740], André 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.

Domaines: Théorie des nombres [math.NT]; Analyse classique [math.CA]

Stephane Fischler, Tanguy Rivoal. Arithmetic theory of E-operators. Journal de l'École polytechnique — Mathématiques, 2016, 3, pp.31-65. ⟨hal-01010964⟩