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

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 ${\bf E}$.