## The zeros of Jensen polynomials are simple

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- by George Csordas and Jack Williamson
- Proc. Amer. Math. Soc.
**49**(1975), 263-264 - DOI: https://doi.org/10.1090/S0002-9939-1975-0361017-4
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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 \[ f(z) = c{z^m}{e^{ - \alpha {z^2} + \beta z}}\coprod \limits _n {(1 - z/{z_n}){e^{z/{z_n}}}} {\text { }},\] 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 \[ {g_n}(x) = \sum \limits _{k = 0}^n {\left ( {\begin {array}{*{20}{c}} n \\ k \\ \end {array} } \right ){a_k}{x^k}} \] 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.## References

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## Bibliographic Information

- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**49**(1975), 263-264 - MSC: Primary 30A08
- DOI: https://doi.org/10.1090/S0002-9939-1975-0361017-4
- MathSciNet review: 0361017