Local analytic solutions of the generalized Dhombres functional equation II

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