Multisummability for some classes of difference equations

Braaksma, Boele L. J.; Faber, Bernard F.

doi:10.5802/aif.1511

Braaksma, Boele L. J. ; Faber, Bernard F.

Annales de l'Institut Fourier, Tome 46 (1996) no. 1, pp. 183-217.

Résumé
Abstract

On considère des équations aux différences finies $y (x + 1) = G (x, y)$ où $G$ prend ses valeurs dans $C^{n}$ et $G$ est méromorphe en $x$ au voisinage de $\infty$ dans $C$ et holomorphe en $y$ au voisinage de 0 dans $C^{n}$ . On montre que sous certaines conditions sur la partie linéaire de $G$ , des solutions formelles dans $C^{n} [[x^{- 1 / p}]], p \in N,$ sont multisommables. De plus on montre que ces solutions formelles se relèvent toujours en des solutions holomorphes dans des demi-plans supérieurs et inférieurs, mais généralement ces solutions ne sont pas déterminées uniquement par les solutions formelles.

This paper concerns difference equations $y (x + 1) = G (x, y)$ where $G$ takes values in $C^{n}$ and $G$ is meromorphic in $x$ in a neighborhood of $\infty$ in $C$ and holomorphic in a neighborhood of 0 in $C^{n}$ . It is shown that under certain conditions on the linear part of $G$ , formal power series solutions in $x^{- 1 / p}, p \in N,$ are multisummable. Moreover, it is shown that formal solutions may always be lifted to holomorphic solutions in upper and lower halfplanes, but in general these solutions are not uniquely determined by the formal solutions.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.1511

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     author = {Braaksma, Boele L. J. and Faber, Bernard F.},
     title = {Multisummability for some classes of difference equations},
     journal = {Annales de l'Institut Fourier},
     pages = {183--217},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {46},
     number = {1},
     year = {1996},
     doi = {10.5802/aif.1511},
     mrnumber = {97e:39002},
     zbl = {0837.39001},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1511/}
}

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Braaksma, Boele L. J.; Faber, Bernard F. Multisummability for some classes of difference equations. Annales de l'Institut Fourier, Tome 46 (1996) no. 1, pp. 183-217. doi : 10.5802/aif.1511. http://www.numdam.org/articles/10.5802/aif.1511/

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