**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*(*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.