Formal solutions of the generalized Dhombres functional equation with value one at zero
type: Item
instance of: scholarly article
title: Formal solutions of the generalized Dhombres functional equation with value one at zero
zbMATH Open document ID: 1243.30059
author: Jörg Tomaschek
author: Ludwig Reich
published in: Aequationes Mathematicae
publication date: 2012-02-22T00:00:00Z
review text: 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)].
reviewed by: Katsuya Ishizaki
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 39B32
Mathematics Subject Classification ID: 30B10
zbMATH DE Number: 6008795
zbMATH Keywords: Dhombres functional equation
zbMATH Keywords: generalized Dhombres functional equation
zbMATH Keywords: solution
MaRDI profile type: Publication
full work available at URL: s00010-011-0104-z
OpenAlex ID: W1973061203
cites work: Q3854084
cites work: Q3058457
cites work: Local analytic solutions of the generalized Dhombres functional equation. II
cites work: On generalized Dhombres equations with nonconstant polynomial solutions in the complex plane
cites work: Analytic solutions of general nonlinear functional equations near resonance
cites work: Small divisor problem for an analytic \(q\)-difference equation
DOI: 10.1007/S00010-011-0104-Z
description: scientific article; zbMATH DE number 6008795