Asymptotics of polynomials orthogonal over circular multiply connected domains

Abstract

Let D be a domain obtained by removing, out of the unit disk {z : |z| < 1}, finitely many mutually disjoint closed disks, and for each integer n ≥ 0, let Pn(z) = z n + · · · be the monic nth-degree polynomial satisfying the planar orthogonality condition´D Pn(z)z m dxdy = 0, 0 ≤ m < n. Under a certain assumption on the domain D, we establish asymptotic expansions and formulae that describe the behavior of Pn(z) as n → ∞ at every point z of the complex plane. We also give an asymptotic expansion for the squared norm´D |Pn| 2 dxdy.