Zeros of meromorphic functions of the form $\sum\limits_n \dfrac{c_n}{(z-t_n)^2}$

TL;DR

This paper investigates the distribution of zeros in a class of meromorphic functions defined by specific sums, establishing a Keldysh-type theorem and exploring links to second-order differential equations with polynomial coefficients.

Contribution

It provides a Keldysh theorem analog for these functions and connects zero-free functions to second-order differential equations with polynomial coefficients.

Findings

Proves an analog of the classical Keldysh theorem for these functions.

Establishes a relationship between zero-free functions and second-order differential equations.

Discusses conditions under which zeros are distributed in these meromorphic functions.

Abstract

We study zeros distribution for meromorphic functions of the form $\sum\limits_n \dfrac{c_n}{(z-t_n)^2}$, where $\sum\limits_n \dfrac{|c_n|}{|t_n|^2} <\infty$. We prove an analog of the classical Keldysh theorem and discuss a relation between zero-free functions of this form and second order differential equtions with polynomial coefficients.