# An example of local analytic q-difference equation : Analytic classification

• [1] UMR 8628, Département de Mathématiques, Université Paris-Sud, Centre d’Orsay, 91405 Orsay Cedex.
• Volume: 15, Issue: 4, page 773-814
• ISSN: 0240-2963

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## Abstract

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Using the techniques developed by Jean Ecalle for the study of nonlinear differential equations, we prove that the $q$-difference equation$x{\sigma }_{q}y=y+b\left(y,x\right)$with $\left({\sigma }_{q}f\right)\left(x\right)=f\left(qx\right)$ ($q>1$) and $b\left(0,0\right)={\partial }_{y}b\left(0,0\right)=0$ is analytically conjugated to one of the following equations :$x{\sigma }_{q}y=y\phantom{\rule{1em}{0ex}}\text{ou}\phantom{\rule{1em}{0ex}}x{\sigma }_{q}y=y+x$

## How to cite

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Menous, Frédéric. "An example of local analytic q-difference equation : Analytic classification." Annales de la faculté des sciences de Toulouse Mathématiques 15.4 (2006): 773-814. <http://eudml.org/doc/10022>.

@article{Menous2006,
abstract = {Using the techniques developed by Jean Ecalle for the study of nonlinear differential equations, we prove that the $q$-difference equation$x \sigma \_q y = y + b ( y, x )$with $( \sigma _q f ) ( x ) = f ( q x )$ ($q &gt; 1$) and $b ( 0, 0 ) = \partial _y b ( 0, 0 ) = 0$ is analytically conjugated to one of the following equations :$x \sigma \_q y = y \quad \text\{ou\}\quad x \sigma \_q y = y + x$},
affiliation = {UMR 8628, Département de Mathématiques, Université Paris-Sud, Centre d’Orsay, 91405 Orsay Cedex.},
author = {Menous, Frédéric},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {nonlinear analytic -difference equations},
language = {eng},
number = {4},
pages = {773-814},
publisher = {Université Paul Sabatier, Toulouse},
title = {An example of local analytic q-difference equation : Analytic classification},
url = {http://eudml.org/doc/10022},
volume = {15},
year = {2006},
}

TY - JOUR
AU - Menous, Frédéric
TI - An example of local analytic q-difference equation : Analytic classification
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2006
PB - Université Paul Sabatier, Toulouse
VL - 15
IS - 4
SP - 773
EP - 814
AB - Using the techniques developed by Jean Ecalle for the study of nonlinear differential equations, we prove that the $q$-difference equation$x \sigma _q y = y + b ( y, x )$with $( \sigma _q f ) ( x ) = f ( q x )$ ($q &gt; 1$) and $b ( 0, 0 ) = \partial _y b ( 0, 0 ) = 0$ is analytically conjugated to one of the following equations :$x \sigma _q y = y \quad \text{ou}\quad x \sigma _q y = y + x$
LA - eng
KW - nonlinear analytic -difference equations
UR - http://eudml.org/doc/10022
ER -

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