A generalization of Laguerre's theorems on zeros of entire functions
Let \(f\), \(\varphi\) be entire functions such that \(f\) is of the Laguerre-Pólya class (i.e. \(f\) can be approximated locally uniformly in \(\mathbb C\) by polynomials having only real zeros) and \(\varphi\) is of the form \(\varphi(w)=\exp(aw)\prod_{n=1}^\infty(1+w/w_{n})\exp(-w/w_{n})\), where \(a\in\mathbb R\), \(w_{n}>0\), and \(\sum_{n=1}^\infty w_{n}^{-2}<+\infty\). Put \((Af)(z):=\sum_{n=0}^\infty\varphi(n)\frac{f^{(n)}(0)}{n!}z^{n}\). The author studies relations between zeros of \(f\), \(\varphi\), and \(Af\).
scientific article; zbMATH DE number 1238027