Siegel's problem for E-functions of order 2

$E$-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 Andr\'e'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 {\em 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 $\ge 3$.