One helpful property of functions generating P\'olya frequency sequences

TL;DR

This paper investigates the zeros of functions generated by Pólya frequency sequences, revealing their simple and evenly distributed zeros, and applies these findings to specific entire functions like the disturbed exponential and partial theta functions.

Contribution

It establishes the zero distribution properties of solutions to a class of equations involving Pólya frequency generating functions, extending understanding of their zero localization and simplicity.

Findings

Zeros are simple or at most double for real parameters.

Zeros are evenly split among specific sectors in the complex plane.

Certain entire functions have simple zeros with distinct absolute values under specific conditions.

Abstract

In this work we study the solutions of the equation $z^pR(z^k)=\alpha$ with nonzero complex $\alpha$, integer $p,k$ and $R(z)$ generating a (possibly doubly infinite) totally positive sequence. It is shown that the zeros of $z^pR(z^k)-\alpha$ are simple (or at most double in the case of real $\alpha^k$) and split evenly among the sectors $\{\frac jk \pi\le\operatorname{Arg} z\le\frac {j+1}k \pi\}$, $j=0,\dots, 2k-1$. Our approach rests on the fact that $z(\ln z^{p/k}R(z) )'$ is an $\mathcal R$-function (i.e. maps the upper half of the complex plane into itself). This result guarantees the same localization to zeros of entire functions $f(z^k)+z^p g(z^k)$ and $g(z^k)+z^{p}f(z^k)$ provided that $f(z)$ and $g(-z)$ have genus $0$ and only negative zeros. As an application, we deduce that functions of the form $\sum_{n=0}^\infty (\pm i)^{n(n-1)/2}a_n z^{n}$ have simple zeros distinct in absolute value under a certain condition on the coefficients $a_n\ge 0$. This includes the "disturbed exponential" function corresponding to $a_n= q^{n(n-1)/2}/n!$ when $0<q\le 1$, as well as the partial theta function corresponding to $a_n= q^{n(n-1)/2}$ when $0<q\le q_*\approx 0.7457224107$.