Preprint MA 62/2007

Local analytic solutions of the generalized Dhombres functional equation II

Ludwig Reich, Jaroslav Smítal, Marta ©tefánková

We study local analytic solutions f of the generalized Dhombres functional equation f(zf(z)) = φ(f(z)), where φ is holomorphic at w0 ≠ 0, f is holomorphic in some open neighborhood of 0, depending on f, and f(0) = w0. After deriving necessary conditions on φ for the existence of nonconstant solutions f with f(0) = w0 we describe, assuming these conditions, the structure of the set of all formal solutions, provided that w0 is not a root of 1. If |w0| ≠ 1 or if w0 is a Siegel number we show that all formal solutions yield local analytic ones. For w0 with 0 < |w0| < 1 we give representations of these solutions involving infinite products.