Local analytic solutions of the generalized Dhombres functional equation II

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.