Local invertible analytic solutions for an iterative differential equation related to a discrete derivatives sequence

In this paper, we are concerned with the existence of analytic solutions of a class of iterative differential equation f (z) = 1 K(f1(z))a1(f2(z))a2 ···(fn(z))an , in the complex field C, where K ∈ C \ {0}, ai ∈ R, fi (z) denotes ith iterate of f (z), i = 1, 2, . . . , n. The above equation is closely related to a discrete derivatives sequence F (m) (see [Y.-F.S. Pétermann, Jean-Luc Rémy, Ilan Vardi, Discrete derivative of sequences, Adv. in Appl. Math. 27 (2001) 562–584]). We first give the existence of analytic solutions of the form of power functions for such an equation. Then by constructing a convergent power series solution y(z) of an auxiliary equation of the form x (z) = Kαx (αz) x(αz) a1 x α2z a2 ··· x αnz an , invertible analytic solutions of the form f (z) = x(αx−1(z)) for the original equation are obtained. We discuss not only the constant α at resonance, i.e. at a root of the unity, but also those α near resonance (near a root of the unity) under the Brjuno condition. © 2007 Elsevier Inc. All rights reserved