Factors of E-operators with an η-apparent singularity at zero
Résumé
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.
Domaines
Théorie des nombres [math.NT]Origine | Fichiers produits par l'(les) auteur(s) |
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