Hypertranscendency of meromorphic solutions of a linear functional equation
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.
scientific article; zbMATH DE number 1295523