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On necessary and sufficient conditions for entire functions to belong to the Laguerre-Pólya class in terms of their Taylor coefficients

  • Thu Hien Nguyen (V.N. Karazin Kharkiv National University)
E1 05 (Leibniz-Saal)

Abstract

We discuss new necessary and sufficient conditions for the entire functions with positive Taylor coefficients to belong to the Laguerre--Pólya class. It is an important class of entire functions which are locally the limit of a sequence of real polynomials having only real zeros. For an entire function $f(z) = \sum_{k=0}^{\infty} a_k z^k,$ we define the second quotients of Taylor coefficients as $q_n(f) := \frac{a_{n-1}^2}{a_{n-2} a_{n}}, n\geq 2$ and find conditions on $q_n(f)$ for $f$ to belong to the Laguerre--Pólya class. Besides, we show the relation of the conditions to the partial theta function $g_a =\sum _{k=0}^{\infty} \frac {z^k}{a^{k^2}}$, when $a>1.$

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Mirke Olschewski

MPI for Mathematics in the Sciences Contact via Mail

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