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

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 $\mathcal{C}$.