2023/01/31 by Stéphane Fischler, Fischler, Stéphane, Tanguy Rivoal +1 · 1 citation
Mathematics · #11J91 (Primary) #33B15 (Secondary) #Advanced Mathematical Identities #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2301.13518
openalex publication_date 2023/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ∞ 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 André and Beukers for E-functions) and the conjecture \bf D∩\bf E=\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.