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

Abstract : In 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, André proved that the finite singularities of a differential operator in Qbar(z)[d/dz] \ {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 η-apparent, in the sense that there exists η ∈ C such that L has a local basis of solutions at 0 in z^ηC[[z]]. We prove that they have a C-basis of solutions of the form Q_j(z)z ^η e^{β_j z}, where η ∈ Q, the β_j ∈ Qbar are pairwise distinct and the Q_j(z) ∈ Qbar[z] \ {0}. This generalizes a previous result by Roques and the author concerning E-operators with an apparent singularity at the origin, of which certain consequences are also given here.
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Contributor : Tanguy Rivoal <>
Submitted on : Wednesday, September 30, 2020 - 10:58:09 AM
Last modification on : Monday, December 14, 2020 - 6:08:24 PM
Long-term archiving on: : Monday, January 4, 2021 - 8:49:13 AM


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  • HAL Id : hal-02953453, version 1



T Rivoal. Factors of E-operators with an η-apparent singularity at zero. 2020. ⟨hal-02953453⟩



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