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

Abstract

In this paper, we are concerned with the existence of analytic solutions of a class of iterative differential equation f ′ ( z ) = 1 K ( f 1 ( z ) ) a 1 ( f 2 ( z ) ) a 2 ⋯ ( f n ( z ) ) a n , in the complex field C , where K ∈ C ∖ { 0 } , a i ∈ R , f i ( 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 ) = K α x ′ ( α z ) ( x ( α z ) ) a 1 ( x ( α 2 z ) ) a 2 ⋯ ( x ( α n z ) ) a n , invertible analytic solutions of the form f ( z ) = x ( α x −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.