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 Hm(z) with generating function. We provide an explicit formula for the limiting pair correlation function of the roots of n m=0 Hm(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.

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

“…If N (t, z) is not a monomial, the root distribution will be different. The quadratic case D(t, z) = 1 + B(z)t + A(z)t 2 is not difficult and it is also mentioned in [13]. We present this case in Section 2 because it gives some directions to our main cases, the cubic and quartic denominators D(t, z), in Sections 3 and 4.…”