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- W3208589802 abstract "E-functions are entire functions with algebraic Taylor coefficients at the origin satisfying certain arithmetic conditions, and solutions of linear differential equations with coefficients in (overline{mathbb Q}(z)); they naturally generalize the exponential function. Siegel and Shidlovsky proved a deep transcendence result for their values at algebraic points. Since then, a lot of work has been devoted to apply their theorem to special E-functions, in particular the hypergeometric ones. In fact, Siegel asked whether any E-function can be expressed as a polynomial in z and generalized confluent hypergeometric series. As a first positive step, Shidlovsky proved that E-functions with order of the differential equation equal to 1 are in (overline{mathbb Q}[z]e^{overline{mathbb Q}z}). In this paper, we give a new proof of a result of Gorelov that any E-function (in the strict sense) with order (le 2) can be written in the form predicted by Siegel with confluent hypergeometric functions ({}_1F_1[alpha ;beta ;lambda z]) for suitable (alpha , beta in mathbb Q) and (lambda in overline{mathbb Q}). Gorelov’s result is in fact more general as it holds for E-functions in the large sense. Our proof makes use of André’s results on the singularities of the minimal differential equations satisfied by E-functions, together with a rigidity criterion for (irregular) differential systems recently obtained by Bloch-Esnault and Arinkin. An ad-hoc version of this criterion had already been used by Katz in his study of confluent hypergeometric series. Siegel’s question remains unanswered for orders (ge 3)." @default.
- W3208589802 created "2021-11-08" @default.
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- W3208589802 date "2021-01-01" @default.
- W3208589802 modified "2023-09-28" @default.
- W3208589802 title "Siegel’s Problem for E-Functions of Order 2" @default.
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- W3208589802 doi "https://doi.org/10.1007/978-3-030-84304-5_21" @default.
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