Factors of E-operators with an η-apparent singularity at zero - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2020

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

T Rivoal
  • Fonction : Auteur

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.
Fichier principal
Vignette du fichier
factorseop.pdf (357.88 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-02953453 , version 1 (30-09-2020)
hal-02953453 , version 2 (26-08-2021)

Identifiants

  • HAL Id : hal-02953453 , version 1

Citer

T Rivoal. Factors of E-operators with an η-apparent singularity at zero. 2020. ⟨hal-02953453v1⟩
45 Consultations
70 Téléchargements

Partager

Gmail Mastodon Facebook X LinkedIn More