Cardon, D. A.; de Gaston, S. A. Differential operators and entire functions with simple real zeros. (English) Zbl 1066.30031 J. Math. Anal. Appl. 301, No. 2, 386-393 (2005). A real entire function \(\varphi(x)\) is said to be in the Laguerre-Pólya class if \(\varphi(x)\) can be expressed in the form \[ \varphi(x)=cx^ne^{-\alpha x^2+\beta x} \prod_{k=1}^\infty\left(1+\frac x{x_k}\right)e^{-\frac x{x_k}}, \] where \(c,\beta, x_k \in \mathbb R\), \(\alpha \geq 0\), \(n\) is a nonnegative integer and \(\sum_{k=1}^\infty 1/x_k^2 < \infty\). Let \(D:=\dfrac d{dx}\). If \(\varphi(x)=\sum_{k=0}^\infty \alpha_kx^k\) and \(f(x)\) are functions in the Laguerre-Pólya class, then under suitable hypotheses (Lemma 2) the function \(\varphi(D)f(x)=\sum_{k=0}^\infty \alpha_kf^{(k)}(x)\) also belongs to the Laguerre-Pólya class.In this well-written paper, the authors solve an open problem concerning the simplicity of the zeros of \(\varphi(D)f(x)\). In particular, they prove the following theorem. Let \(\varphi\) and \(f\) be functions in the Laguerre-Pólya class. Set \(\varphi(x)=e^{-\alpha x^2}\varphi_1(x)\) and \(f(x)= e^{-\beta x^2}f_1(x)\), where \(\varphi_1\) and \(f_1\) have genus \(0\) or \(1\), and \(\alpha, \beta\geq 0\). If \(\alpha \beta < 1/4\) and if \(\varphi\) has infinitely many zeros, then \(\varphi(D)f(x)\) has only simple real zeros. Reviewer: George Csordas (Honolulu) Cited in 3 Documents MSC: 30D15 Special classes of entire functions of one complex variable and growth estimates 30C10 Polynomials and rational functions of one complex variable Keywords:differential operators; Laguerre-Pólya class; simplicity of zeros PDFBibTeX XMLCite \textit{D. A. Cardon} and \textit{S. A. de Gaston}, J. Math. Anal. Appl. 301, No. 2, 386--393 (2005; Zbl 1066.30031) Full Text: DOI References: [1] Craven, T.; Csordas, G., Differential operators of infinite order and the distribution of zeros of entire functions, J. Math. Anal. Appl., 186, 799-820 (1994) · Zbl 0810.30006 [2] Levin, B. Ja., Distribution of Zeros of Entire Functions (1980), American Mathematical Society: American Mathematical Society Providence, RI [3] Skovgaard, H., On inequalities of the Turán type, Math. Scand., 2, 65-73 (1954) · Zbl 0055.29904 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.