Asymptotics of polynomials orthogonal over circular multiply connected domains

Let $D$ be a domain obtained by removing, out of the unit disk $\{z:\mathrm{\mid }z\mathrm{\mid }<1\}$, finitely many mutually disjoint closed disks, and for each integer $n\ge 0$, let $P_n(z)=z^n+\cdots$ be the monic $n$th-degree polynomial satisfying the planar orthogonality condition ${\int }_D{P}_n(z)\overline{z^m}dxdy=0$, $0\leq m<n$\mathrm{.}Underacertainassumptiononthedomain$D$, we establish asymptotic expansions and formulae that describe the behavior of $P_n(z)$ as $n\to \mathrm{\infty }$ at every point $z$ of the complex plane. We also give an asymptotic expansion for the squared norm ${\int }_D\mathrm{\mid }{P}_n{\mathrm{\mid }}^{2}dxdy$.