Some remarks to the formal and local theory of the generalized Dhombres functional equation (Q1936932)

The authors consider local analytic formal solutions of the generalized Dhombres functional equation \[ f(zf(z))=\varphi(f(z)) \] in the complex domain, where \(\varphi(z)\) is a known function. Let \(w_0\) be a complex number different from zero and no root of unity. If \(f(0)=w_0\) and \(f(z)\) is not a constant, then we write \(f(z)=w_0+g(z)\), \(g(z)=c_kz^k+c_{k+1}z^{k+1}+\dots\), \(k\in \mathbb N\), \(c_k\neq0\). Then (1) is transformed into the generalized Dhombers functional equation \[ g(w_0z+zg(z))=\tilde\varphi(g(z)), \] where \(\varphi(y)=w_0+\tilde\varphi(y-w_0)\). Denote by \(\Gamma_1\) the set of all formal series beginning with \(z\). It is shown that there exists a unique function \(\tilde {g_0}\in \Gamma_1\) such that the set of non-constant solutions \(g\) of (2) in \(\mathbb {C} [z]\) is given by \[ {\mathcal L}_{\tilde\varphi}\{g|g(z)=\tilde g_0(c_kz^k), c_k\in\mathbb C^*\}. \] Then the set \({\mathcal L}_{\tilde\varphi}\) is a subset of \(\mathbb {C}[z^k]\). Furthermore, (2) has a unique solution \(g\) such that \(g(z)=c_kz^k+\dots\), for every \(c_k\in\mathbb C^*\). The authors also represent solutions of (2) by means of an infinite product.