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

摘要

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 Π~n _m=_0H_m(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.