Factors of $E$-operators with an $\eta$-apparent singularity at zero

Abstract

In 1929, Siegel defined $E$-functions as power series in $\overline{\mathbb{Q}}[[z]]$, with Taylor coefficients satisfying certain growth conditions, and solutions of linear differential equations with coefficients in $\overline{\mathbb{Q}}(z)$. The Siegel–Shidlovskii theorem (1956) generalized to $E$-functions the Diophantine properties of the exponential function. In 2000, André proved that the finite singularities of a differential operator in $\overline{\mathbb{Q}}(z)[d/dz] \setminus \{0\}$ of minimal order for some non-zero $E$-function are apparent, except possibly 0 which is always regular singular. We pursue the classification of such operators and consider those for which 0 is $\eta$-apparent, in the sense that there exists $\eta \in \mathbb C$ such that $L$ has a local basis of solutions at 0 in $z^{\eta} \mathbb C[[z]]$. We prove that they have a $\mathbb{C}$-basis of solutions of the form $Q_{j}(z)z^{\eta} e^{\beta_{j} z}$, where $\eta \in \mathbb{Q}$, the $\beta_{j} \in \overline{\mathbb{Q}}$ are pairwise distinct and the $Q_{j}(z) \in \overline{\mathbb{Q}}[z] \setminus \{0\}$. This generalizes a previous result by Roques and the author concerning $E$-operators with an apparent singularity or no singularity at the origin, of which certain consequences are also given here.