Formal solutions of the generalized Dhombres functional equation with value one at zero
The authors continue the research on the Dhombres functional equation \[ f(zf(z))=\varphi(f(z))\tag{1} \] in a complex domain, where \(\varphi(z)\) is known. See, e.g., [\textit{L. Reich} and \textit{J. Smítal}, J. Difference Equ. Appl. 15, No. 11--12, 1179--1191 (2009; Zbl 1178.39034)] for additional information on problems related to the Dhombres equation. Set \(f(z)=w_0+g(z)\) in (1), where \(g(z)=c_kz^k+c_{k+1}z^{k+1}+\dots\), which leads to the transformed generalized Dhombres functional equation \(g(w_0z+zg(z))=\tilde{\varphi}(g(z))\). There are already several results for the situation \(f(0) = w_{0}\) and \(w_0 \in {\mathbb{C}} \setminus {\mathbb{E}}\), where \({\mathbb{E}}\) denotes the complex roots of \(1\). In this paper, the case \(f(0)=1\) is discussed. Using new methods, the authors obtain necessary and sufficient conditions on \(\tilde{\varphi}\) for the existence of a non-constant formal solution of (1) with \(f(0)=1\). This is an improvement of the result in [\textit{L. Reich} and \textit{J. Smítal}, Aequationes Math. 80, No. 1--2, 201--208 (2010; Zbl 1210.39026)].
generalized Dhombres functional equation
scientific article; zbMATH DE number 6008795