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

中文翻译：

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在圆形多重连接域上正交的多项式的渐近性

让 $\mathcal{d}$ 是通过从单位磁盘中删除而获得的域 $\{ž：\left|ž\right|<\mathrm{1个}\}$，有限多个相互不相交的封闭磁盘，并且对于每个整数 $ñ\ge 0$，让 $P_{ñ}（ž）={ž}^{ñ}+\cdots$ 成为莫妮卡 $ñ$满足平面正交性条件的三次多项式 ${\int }_{\mathcal{d}}{P}_{ñ}（ž）\overline{{ž}^{米}}dXdÿ=0$， $0\le 米<ñ$。在域的一定假设下$\mathcal{d}$，我们建立了渐近展开和公式来描述行为 $P_{ñ}（ž）$ 如 $ñ\to \infty$ 在每一点上 $ž$复杂平面的 我们还给出平方范数的渐近展开${\int }_{\mathcal{d}}{\left|P_{ñ}\right|}^{2}dXdÿ$。