Relations between values of arithmetic Gevrey series, and applications to values of the Gamma function

TL;DR

This paper explores the relationships between special classes of arithmetic Gevrey series and their values at algebraic points, leading to new results on the transcendence and algebraic independence of important mathematical constants.

Contribution

It establishes that elements of the class ${f G}$ can be expressed via polynomials in classes ${f E}$ and ${f D}$, introduces mixed functions, and derives several Diophantine and transcendence results.

Findings

Values of the Gamma function and its derivatives at non-integer algebraic points are transcendental.

Euler's constant is not in the class ${f E}$.

The paper proves an analogue of Beukers' linear independence theorem for mixed functions.

Abstract

We investigate the relations between the rings ${\bf E}$, ${\bf G}$ and ${\bf D}$ of values taken at algebraic points by arithmetic Gevrey series of order either $-1$ ($E$-functions), $0$ (analytic continuations of $G$-functions) or $1$ (renormalization of divergent series solutions at $\infty$ of $E$-operators) respectively. We prove in particular that any element of ${\bf G}$ can be written as multivariate polynomial with algebraic coefficients in elements of ${\bf E}$ and ${\bf D}$, and is the limit at infinity of some $E$-function along some direction. This prompts to defining and studying the notion of mixed functions, which generalizes simultaneously $E$-functions and arithmetic Gevrey series of order 1. Using natural conjectures for arithmetic Gevrey series of order 1 and mixed functions (which are analogues of a theorem of Andr\'e and Beukers for $E$-functions) and the conjecture ${\bf D}\cap{\bf E}=\overline{\mathbb Q}$ (but not necessarily all these conjectures at the same time), we deduce a number of interesting Diophantine results such as an analogue for mixed functions of Beukers' linear independence theorem for values of $E$-functions, the transcendance of the values of the Gamma function and its derivatives at all non-integral algebraic numbers, the transcendance of Gompertz constant as well as the fact that Euler's constant is not in ${\bf E}$.