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• W4225272038 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." @default.
• W4225272038 created "2022-05-04" @default.
• W4225272038 creator A5010336542 @default.
• W4225272038 date "2022-07-22" @default.
• W4225272038 modified "2023-09-25" @default.
• W4225272038 title "Factors of $E$-operators with an $eta$-apparent singularity at zero" @default.
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• W4225272038 doi "https://doi.org/10.2969/jmsj/85708570" @default.
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