Hypertranscendency of meromorphic solutions of a linear functional equation
type: Item
instance of: scholarly article
title: Hypertranscendency of meromorphic solutions of a linear functional equation
zbMATH Open document ID: 0932.39020
author: Katsuya Ishizaki
published in: Aequationes Mathematicae
publication date: 2000-03-06T00:00:00Z
review text: The author considers meromorphic solutions \(y\) of the functional equation \(y(cz)=a(z)y(z)+b(z)\) where \(a\) and \(b\) are meromorphic and \(c\) is a non-zero constant satisfying \(|c|\not=1\). The first result says that if \(b(0)\not=\infty, a(0)\not=0, \infty\) and \(a(0)\not=c^{j}\) for \(j=0,1,2,...,\) then the functional equation has a meromorphic solution. A second result says that if \(a\) and \(b\) are rational functions, then all transcendental solutions of the functional equation are hypertranscendental; that is, they do not satisfy an algebraic differential equation. These results extend previous work of \textit{H. Wittich} [Arch. Math. 2, 90-95 (1950; Zbl 0037.33402)] dealing with the case that \(a\) and \(b\) are polynomials and \(y\) is entire.
Mathematics Subject Classification ID: 39B32
Mathematics Subject Classification ID: 30D05
zbMATH DE Number: 1295523
zbMATH Keywords: linear functional equation
zbMATH Keywords: hypertranscendency
zbMATH Keywords: meromorphic solutions
zbMATH Keywords: transcendental solutions
reviewed by: Walter Bergweiler
MaRDI profile type: Publication
full work available at URL: s000100050062
OpenAlex ID: W2014311669
DOI: 10.1007/S000100050062
description: scientific article; zbMATH DE number 1295523