no. 1
Accelero-summation of the formal solutions of nonlinear difference equations
[Accéléro-sommation des solutions formelles d’équations aux différences nonlinéaires]
Immink, Geertrui Klara1
1 University of Groningen Faculty of Economics P.O. Box 800 9700 AV Groningen (The Netherlands)
Annales de l'Institut Fourier, Tome 61 (2011) no. 1, pp. 1-51.

En 1996, Braaksma et Faber ont établi la multi-sommabilité, sur des multi-intervalles convenables, des solutions formelles d’équations aux différences nonlinéaires, localement analytiques, sous la condition que le niveau 1 + ne se présente pas. En combinant leurs résultats avec d’autres récents pour le cas des deux niveaux 1 et 1 + , on démontre, pour une classe très générale d’équations, l’accéléro-sommabilité de la solution formelle. L’accéléro-somme est solution analytique de l’équation, admettant la solution formelle comme développement asymptotique à l’infini.

In 1996, Braaksma and Faber established the multi-summability, on suitable multi-intervals, of formal power series solutions of locally analytic, nonlinear difference equations, in the absence of “level 1 + ”. Combining their approach, which is based on the study of corresponding convolution equations, with recent results on the existence of flat (quasi-function) solutions in a particular type of domains, we prove that, under very general conditions, the formal solution is accelero-summable. Its sum is an analytic solution of the equation, represented asymptotically by the formal solution in a certain unbounded domain.

DOI : 10.5802/aif.2596
Classification : 39A10, 30E15, 40G10
Keywords: Nonlinear difference equation, formal solution, accelero-summation, quasi-function
Mot clés : équation aux différences nonlinéaire, solution formelle, accéléro-sommation, quasi-fonction

Immink, Geertrui Klara 1

1 University of Groningen Faculty of Economics P.O. Box 800 9700 AV Groningen (The Netherlands) 
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Immink, Geertrui Klara. Accelero-summation of the formal solutions of nonlinear difference equations. Annales de l'Institut Fourier, Tome 61 (2011) no. 1, pp. 1-51. doi : 10.5802/aif.2596. https://aif.centre-mersenne.org/articles/10.5802/aif.2596/

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