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Pair correlation of roots of rational functions with rational generating functions and quadratic denominators
Authors:Khang Tran  Alexandru Zaharescu
Institution:1. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL, 61801, USA
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 ∑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→∞, 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 $\mathcal{C}$ .
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