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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The zeros of Jensen polynomials are simple


Authors: George Csordas and Jack Williamson
Journal: Proc. Amer. Math. Soc. 49 (1975), 263-264
MSC: Primary 30A08
MathSciNet review: 0361017
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Abstract: An entire function $ f(z) = \Sigma _{k = 0}^\infty {a_k}{z^{k + m}}/k!$ is said to be in the class $ \mathcal{L} - \mathcal{P}$ (Laguerre-Pólya) if it can be represented in the form

$\displaystyle f(z) = c{z^m}{e^{ - \alpha {z^2} + \beta z}}\coprod\limits_n {(1 - z/{z_n}){e^{z/{z_n}}}} {\text{ }},$

where $ \alpha \geq 0,c,\beta $ and $ {z_n}$ are real, and $ {\Sigma _n}z_n^{ - 2} < \infty $. A well-known result of Jensen asserts that the associated (Jensen) polynomials

$\displaystyle {g_n}(x) = \sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}{c}} n \\ k \\ \end{array} } \right){a_k}{x^k}} $

have only real zeros. Here we present an elementary proof of this fact; we also show that the zeros of $ {g_n}(x)$ are simple.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0361017-4
PII: S 0002-9939(1975)0361017-4
Keywords: Jensen polynomials, real, simple zeros, entire function
Article copyright: © Copyright 1975 American Mathematical Society