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 equationf′(z)=1K(f1(z))a1(f2(z))a2⋯(fn(z))an, in the complex field CC, where K∈C∖{0}K∈C∖{0}, ai∈Rai∈R, fi(z)fi(z) denotes i  th iterate of f(z)f(z), i=1,2,…,ni=1,2,…,n. The above equation is closely related to a discrete derivatives sequence F′(m)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)y(z) of an auxiliary equation of the formx′(z)=Kαx′(αz)(x(αz)a1)(x(α2z)a2)⋯(x(αnz)an),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))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.