Some remarks to the formal and local theory of the generalized Dhombres functional equation

We are looking for local analytic respectively formal solutions of the generalized Dhombres
functional equation $f(zf(z))=\varphi(f(z))$ in the complex domain. First we give two
proofs of the existence theorem about solutions $f$ with $f(0) = w_0$ and $w_0 \in
\mathbb{C}^\star \setminus \mathbb{E}$ where $\mathbb{E}$ denotes the group of complex roots of $1$. Afterwards we represent solutions $f$ by means
of infinite products where we use on the one hand the canonical convergence of complex
analysis, on the other hand we show how solutions converge with respect to the weak
topology. In this section we also study solutions where the initial value $z_0$ is
different from zero.