Local analytic solutions of the generalized Dhombres functional equation. II (Q1022985)

The authors considered the set of local analytic solutions of the generalized Dhombres functional equation \[ f(zf(z))= \varphi(f(z))\tag{1} \] with \(f(0)=0\) in part I [Sitzungsber., Abt. II, Österr. Akad. Wiss., Math.-Naturwiss. Kl. 214, 3--25 (2005; Zbl 1117.39015)]. Now they study the case \(f(0) \neq 0\). Let \(w_{0} \in \mathbb C^{*}\) and let \(\varphi\) be an analytic function in a neighbourhood of \(w_{0}\). If the functional equation (1) has a nonconstant local analytic solution \(f\) in a neighbourhood of zero with \[ f(0) = w_{0}, \] then there exists \(k \in \mathbb N\) such that \[ \text{ord}(f(z) - w_{0}) = k \] and \[ \varphi(w) = w_{0} + w_{0}^{k}(w - w_{0}) + d_{2}(w - w_{0})^{2} +\dots\tag{2} \] Moreover, if \(w_{0}\) is not a root of \(1\), then \(k\) is uniquely determined. Conversely, if \(k \in \mathbb N, w_{0} \in \mathbb C\) is a Siegel number or \(|w_{0}| \neq 1\) and \(\varphi\) is a holomorphic function of the form (2) in a neighbourhood of \(w_{0}\), then the set of all nonconstant local analytic solutions of (1) with \(f(0) = w_{0}\) is given by \[ \{ f(z) = w_{0} + \tilde{g}_{0}(c_{k}z^{k}): c_{k} \in \mathbb C^{*} \}, \] where \(\tilde{g}_{0}\) is a unique convergent power series such that \(\text{ord}(\tilde{g}_{0}(y) - y) \geq 2\).