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Differential operators and entire functions with simple real zeros. (English) Zbl 1066.30031

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.

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
30C10 Polynomials and rational functions of one complex variable
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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
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