# Asymptotics of polynomials orthogonal over circular multiply connected domains

## TL;DR

This paper investigates the asymptotic behavior of orthogonal polynomials over complex domains formed by removing finitely many disks from the unit disk, providing detailed expansions and formulas as the degree grows large.

## Contribution

It establishes new asymptotic expansions and formulas for orthogonal polynomials over multiply connected domains, extending classical results to more complex geometries.

## Key findings

- Asymptotic expansions for $P_n(z)$ as $n o
$
- Explicit formulas for the squared norm of $P_n$
- Behavior characterized at every point in the complex plane

## 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 $n\geq 0$, let $P_n(z)=z^n+\cdots$ be the monic $n$th-degree polynomial satisfying the planar orthogonality condition $\int_D P_n(z)\overline{z^m}dxdy=0$, $0\leq m<n$. Under a certain assumption on the domain $D$, we establish asymptotic expansions and formulae that describe the behavior of $P_n(z)$ as $n\to\infty$ at every point $z$ of the complex plane. We also give an asymptotic expansion for the squared norm $\int_D|P_n|^2dxdy$.