Some results on entire functions from the Laguerre-Pólya class: proof ideas and techniques

Abstract: The Laguerre-P\'olya class is a class of entire functions that are locally the uniform limit of a sequence of real polynomials that have only real zeros. We present some simple necessary and sufficient conditions for entire functions to belong to the Laguerre–Pólya class in terms of their Taylor coefficients. 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--P\'olya class, or to have only real zeros. In this talk, we will focus on the entire functions with increasing second quotients of Taylor coefficients, and discuss proof ideas and techniques we used. This is joint work with Anna Vishnyakova.

complex variablesdynamical systems

Audience: researchers in the topic

CAvid: Complex Analysis video seminar

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