Sommation effective d’une somme de Borel par séries de factorielles

Delabaere, Eric; Rasoamanana, Jean-Marc

Annales de l'Institut Fourier, Tome 57 (2007) no. 2, pp. 421-456.

Nous abordons dans cet article la question de la sommation effective d’une somme de Borel d’une série par la série de factorielles associée. Notre approche fournit un contrôle de l’erreur entre la somme de Borel recherchée et les sommes partielles de la série de factorielles. Nous généralisons ensuite cette méthode au cadre des séries de puissances fractionnaires, après avoir démontré un analogue d’un théorème de Nevanlinna de sommation de Borel fine pour ce cadre.

In this article, we consider the effective resummation of a Borel sum by its associated factorial series expansion. Our approach provides concrete estimates for the remainder term when truncating this factorial series. We then generalize a theorem of Nevanlinna which gives us the natural framework to extend the factorial series method for Borel-resummable fractional power series expansions.

Classification : 30E15, 40Gxx
Mot clés : sommation de Borel, séries de factorielles.
Keywords: Borel-resummation, factorial series.

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