Factors of E-operators with an η-apparent singularity at zero

International audienceIn 1929, Siegel defined $E$-functions as power series in $\Qbar[[z]]$, with Taylor coefficients satisfying certain growth conditions, and solutions of linear differential equations with coefficients in $\Qbar(z)$. The Siegel-Shidlovskii Theorem (1956) generalized to $E$-functions the Diophantine properties of the exponential function. In 2000, Andr\'e proved that the finite singularities of a differential operator in $\Qbar(z)[\frac{d}{dz}]\setminus\{0\}$ 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 $\beta_j\in \Qbar$ are pairwise distinct and the $Q_j(z)\in \Qbar[z]\setminus\{0\}$. 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