Authors: Augustin Fruchard ^1,2; Reinhard Schäfke ³

• 1 Laboratoire de Mathématiques Informatique et Applications
• 2 Laboratoire de Mathématiques Informatique et Applications [UHA]
• 3 Institut de Recherche Mathématique Avancée [IRMA]

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.