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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Derivatives of entire functions and a question of Pólya


Authors: Simon Hellerstein and Jack Williamson
Journal: Trans. Amer. Math. Soc. 227 (1977), 227-249
MSC: Primary 30A64
DOI: https://doi.org/10.1090/S0002-9947-1977-0435393-4
MathSciNet review: 0435393
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Abstract: An old question of Pólya asks whether an entire function f which has, along with each of its derivatives, only real zeros must be of the form

$\displaystyle f(z) = {z^m}{e^{ - a{z^2} + bz + c}}\prod\limits_n {\left( {1 - \frac{z}{{{z_n}}}} \right)} {e^{z/{z_n}}}$

where $ a \geqslant 0,b$ and the $ {z_n}$ are real, and $ {\Sigma _n}z_n^{ - 2} < \infty $. This note answers this question (essentially in the affirmative) if f is of finite order; indeed, it is established that if $ f,f'$, and $ f''$ have only real zeros (f of finite order), then either f has the above form or f has one of the forms

$\displaystyle f(z) = a{e^{bz}},\quad f(z) = a({e^{icz}} - {e^{id}})$

where a, b, c, and d are constants, b complex, c and d real.

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DOI: https://doi.org/10.1090/S0002-9947-1977-0435393-4
Keywords: Real entire function, finite order, finite genus, derivative
Article copyright: © Copyright 1977 American Mathematical Society