Orthogonal polynomials in weighted Bergman spaces

Abstract

Let $w$ be a weight on the unit disk $\mathbb{D}$ having the form \[w(z)=|v(z)|^2\prod_{k=1}^s\left|\frac{z-a_k}{1-z\overline{a}_k}\right|^{m_k}\,,\quad m_k>-2,\ |a_k|<1,\] where $v$ is analytic and free of zeros in $\overline{\mathbb{D}}$, and let $(p_n{)}_{n=0}^{\infty }$ be the sequence of polynomials ($p_n$ of degree $n$) orthonormal over $\mathbb{D}$ with respect to $w$. We give an integral representation for $p_n$ from which it is in principle possible to derive its asymptotic behavior as $n\to \infty$ at every point $z$ of the complex plane, the asymptotic analysis of the integral being primarily dependent on the nature of the first singularities encountered by the function $v(z{)}^{-1}{\prod }_{k=1}^s(1-z{\overline{a}}_k{)}^{-1}$.