Necessary and sufficient conditions for entire functions to belong to the Laguerre-Polya class

Abstract: The famous Laguerre-Polya class consists of entire functions which are uniform on the compacts limits of real polynomials having all real zeros. The Laguerre-Polya class is of interest to many areas of mathematics such as complex analysis, statistical physics, combinatorics, asymptotic analysis, the theory of mock modular forms and others. We present new necessary and new sufficient conditions for an entire function to belong to the Laguerre-Polya class in terms of Taylor coefficients of the function. The partial theta-function $g_a(z) =\sum_{k=0}^{\infty} \frac {z^k}{a^{k^2}}, a>1,$ plays an important role in our investigations. It is known that there exists a constant $ q_\infty\approx 3{.}23363666,$ such that the partial theta-function belongs to the Laguerre-Polya class if and only if $a^2 \geq q_\infty.$ The following statement is an example of our results. Let $f(z)=\sum_{k=0}^\infty a_k z^k $ be an entire function with positive coefficients. Suppose that the sequence $\frac{a_n^2}{a_{n-1} a_{n+1}}$ is decreasing in $n$, and the limit of this sequence is greater than or equal to $\ q_\infty.$ Then the function $f$ belongs to the Laguerre-Polya class.

complex variablesdynamical systems

Audience: researchers in the topic

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