arima:1986 -
Revue Africaine de Recherche en Informatique et Mathématiques Appliquées,
August 1, 2015,
Volume 20 - 2015 - Special issue - Colloquium in Honor of Éric Benoît
-
https://doi.org/10.46298/arima.1986
Composite Asymptotic Expansions and Difference Equations
1 Laboratoire de Mathématiques Informatique et Applications
2 Laboratoire de Mathématiques Informatique et Applications [UHA]
3 Institut de Recherche Mathématique Avancée
Difference equations in the complex domain of the form y(x+ϵ)−y(x)=ϵf(y(x))/y(x) are considered. The step size ϵ>0 is a small parameter, and the equation has a singularity at y=0. Solutions near the singularity are described using composite asymptotic expansions. More precisely, it is shown that the derivative v′ of the inverse function v of a solution (the so-called Fatou coordinate) admits a Gevrey asymptotic expansion in powers of the square root of ϵ, denoted by η, involving functions of y and of Y=y/η. This also yields Gevrey asymptotic expansions of the so-called Écalle-Voronin invariants of the equation which are functions of epsilon. An application coming from the theory of complex iteration is presented.