Pair correlation of roots of rational functions with rational generating functions and quadratic denominators

TL;DR

This paper derives an explicit formula for the limiting pair correlation of roots of products of rational functions with quadratic denominators, revealing detailed root distribution behavior as the degree grows large.

Contribution

It provides a new explicit formula for the limiting pair correlation function of roots of certain rational functions with quadratic generating functions.

Findings

Explicit formula for pair correlation function derived

Root distribution analyzed on specific subarcs of a curve

Example showing non-existence of correlation function at endpoints

Abstract

For any rational functions with complex coefficients $A(z), B(z)$ and $C(z)$, where $A(z)$, $C(z)$ are not identically zero, we consider the sequence of rational functions $H_{m}(z)$ with generating function $\sum H_{m}(z)t^{m}=1/(A(z)t^{2}+B(z)t+C(z))$. We provide an explicit formula for the limiting pair correlation function of the roots of $\prod_{m=0}^{n}H_{m}(z)$, as $n\rightarrow\infty$, counting multiplicities, on certain closed subarcs $J$ of a curve $\mathcal{C}$ where the roots lie. We give an example where the limiting pair correlation function does not exist if $J$ contains the endpoints of $\mathcal{C}$.