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Microsolutions of differential operators and values of arithmetic Gevrey series. (English) Zbl 1431.11093
Summary: We continue our investigation of $$E$$-operators, in particular their connection with $$G$$-operators; these differential operators are fundamental in understanding the diophantine properties of Siegel’s $$E$$ and $$G$$-functions. We study in detail microsolutions (in Kashiwara’s sense) of Fuchsian differential operators, and apply this to the construction of bases of solutions at 0 and $$\infty$$ of any $$E$$-operator from microsolutions of a $$G$$-operator; this provides a constructive proof of a theorem of Y. André [Ann. Math. (2) 151, No. 2, 705–740 (2000; Zbl 1037.11049)]. We also focus on the arithmetic nature of connection constants and Stokes constants between different bases of solutions of $$E$$-operators. For this, we introduce and study in details an arithmetic (inverse) Laplace transform that enables one to get rid of transcendental numbers inherent to André’s original approach. As an application, we define a set of special values of arithmetic Gevrey series, and discuss its conjectural relation with the ring of exponential periods of Kontsevich-Zagier.

##### MSC:
 11J91 Transcendence theory of other special functions 11J81 Transcendence (general theory) 32C38 Sheaves of differential operators and their modules, $$D$$-modules 33E30 Other functions coming from differential, difference and integral equations
##### Keywords:
Gevrey series; exponential periods
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