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

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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 Hₘ(z) with generating function ∑ Hₘ(z)tᵐ=1/(A(z)t²+B(z)t+C(z)). We provide an explicit formula for the limiting pair correlation function of the roots of ∏ₘ₌₀ⁿHₘ(z), as n→∞, counting multiplicities, on certain closed subarcs J of a curve C where the roots lie. We give an example where the limiting pair correlation function does not exist if J contains the endpoints of C.