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

Stéphane Fischler (1) , Tanguy Rivoal (2)

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

We investigate the relations between the rings E, G and 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 infinity of E-operators) respectively. We prove in particular that any element of G can be written as multivariate polynomial with algebraic coefficients in elements of E and 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 André and Beukers for E-functions) and the conjecture that intersection of D and E is reduced to algebraic numbers (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 E.

Domaines: Théorie des nombres [math.NT]

Stéphane Fischler, Tanguy Rivoal. Relations between values of arithmetic Gevrey series, and applications to values of the Gamma function. 2023. ⟨hal-03920630⟩