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 $n\to \infty$, 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 $\mathcal{C}$.

References in corpus (1)

• Discriminants of Polynomials Related to Chebyshev Polynomials: The 'Mutt and Jeff' Syndrome

Cited by in corpus (1)

• Connections between discriminants and the root distribution of polynomials with rational generating function