Accelero-summation of the formal solutions of nonlinear difference equations
Annales de l'Institut Fourier, Volume 61 (2011) no. 1, p. 1-51
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.
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.
DOI : https://doi.org/10.5802/aif.2596
Classification:  39A10,  30E15,  40G10
Keywords: Nonlinear difference equation, formal solution, accelero-summation, quasi-function
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     author = {Immink, Geertrui Klara},
     title = {Accelero-summation of the formal solutions of nonlinear difference equations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {1},
     year = {2011},
     pages = {1-51},
     doi = {10.5802/aif.2596},
     zbl = {1225.39005},
     mrnumber = {2828125},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2011__61_1_1_0}
}
Immink, Geertrui Klara. Accelero-summation of the formal solutions of nonlinear difference equations. Annales de l'Institut Fourier, Volume 61 (2011) no. 1, pp. 1-51. doi : 10.5802/aif.2596. https://aif.centre-mersenne.org/item/AIF_2011__61_1_1_0/

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