Multisummability for some classes of difference equations

Boele L. J. Braaksma; Bernard F. Faber

Annales de l'institut Fourier (1996)

  • Volume: 46, Issue: 1, page 183-217
  • ISSN: 0373-0956

Abstract

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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 ∞ 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 ∈ 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.

How to cite

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Braaksma, Boele L. J., and Faber, Bernard F.. "Multisummability for some classes of difference equations." Annales de l'institut Fourier 46.1 (1996): 183-217. <http://eudml.org/doc/75170>.

@article{Braaksma1996,
abstract = {This paper concerns difference equations $y(x+1)=G(x,y)$ where $G$ takes values in $\{\bf C\}^n$ and $G$ is meromorphic in $x$ in a neighborhood of $\infty $ in $\{\bf C\}$ and holomorphic in a neighborhood of 0 in $\{\bf C\}^n$. It is shown that under certain conditions on the linear part of $G$, formal power series solutions in $x^\{-1/p\}, p\in \{\bf N\},$ 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.},
author = {Braaksma, Boele L. J., Faber, Bernard F.},
journal = {Annales de l'institut Fourier},
keywords = {multisummability; normal forms; Borel and Laplace transforms; Gevrey series; Stokes phenomenon; difference equations; formal power series solutions; holomorphic solutions},
language = {eng},
number = {1},
pages = {183-217},
publisher = {Association des Annales de l'Institut Fourier},
title = {Multisummability for some classes of difference equations},
url = {http://eudml.org/doc/75170},
volume = {46},
year = {1996},
}

TY - JOUR
AU - Braaksma, Boele L. J.
AU - Faber, Bernard F.
TI - Multisummability for some classes of difference equations
JO - Annales de l'institut Fourier
PY - 1996
PB - Association des Annales de l'Institut Fourier
VL - 46
IS - 1
SP - 183
EP - 217
AB - This paper concerns difference equations $y(x+1)=G(x,y)$ where $G$ takes values in ${\bf C}^n$ and $G$ is meromorphic in $x$ in a neighborhood of $\infty $ in ${\bf C}$ and holomorphic in a neighborhood of 0 in ${\bf C}^n$. It is shown that under certain conditions on the linear part of $G$, formal power series solutions in $x^{-1/p}, p\in {\bf N},$ 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.
LA - eng
KW - multisummability; normal forms; Borel and Laplace transforms; Gevrey series; Stokes phenomenon; difference equations; formal power series solutions; holomorphic solutions
UR - http://eudml.org/doc/75170
ER -

References

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