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 $a > 1.$

Seminar on Nonlinear Algebra Seminar on Nonlinear Algebra

MPI for Mathematics in the Sciences G3 10 (Lecture hall)

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Thursday, 19.06.25 Squared Linear Models with Hannah Friedman
Thursday, 19.06.25 Positivity of Schubert coefficients with Colleen Robichaux

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

Abstract

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Seminar on Nonlinear Algebra Seminar on Nonlinear Algebra

Upcoming Events of this Seminar