On the relation between the Borel sum and the classical solution of the Cauchy problem for certain partial differential equations

Abstract

On the relation between the Borel sum and the classical solution of the Cauchy problem for certain partial differential equations Annales de la faculté des sciences de Toulouse 6 e série, tome 14, n o 3 (2005), p. 435-458 <http://www.numdam.org/item?id=AFST_2005_6_14_3_435_0> © Université Paul Sabatier, 2005, tous droits réservés. L'accès aux archives de la revue « Annales de la faculté des sciences de Toulouse » (http://picard.ups-tlse.fr/~annales/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

A formal power series expansion–regularization approach for Lévy stable distributions: the symmetric case with $\alpha =2/M$ (Mpositive integer)

“…There are different techniques to implement this protocol, such as Abelʼs, Borelʼs, Cesàroʼs, Dirichletʼs, Riemannʼs, etc (for a nice didactic review, with many examples, see [37]). For GHFs, the Borel summability method [24,25] is the most used one [18][19][20].…”

“…Divergent F p q ʼs belong to this category of welldefined mathematical objects. (iv) Because the relevance of the F p q functions, there is a large literature analyzing different regularization methods for diverging hypergeometric series [16][17][18][19][20][21][22]. They are mainly based on summability theory [23], especially Borelʼs [24,25].…”