Siegel’s Problem for E-Functions of Order 2

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 $\ge 3$.