Orthogonal polynomials in weighted Bergman spaces

TL;DR

This paper studies orthogonal polynomials in weighted Bergman spaces with specific weights, providing an integral representation that enables analysis of their asymptotic behavior as the degree grows large.

Contribution

It introduces an integral formula for orthogonal polynomials in weighted Bergman spaces with complex weights, facilitating asymptotic analysis at any point in the complex plane.

Findings

Integral representation for orthogonal polynomials derived.

Asymptotic behavior linked to singularities of the weight function.

Potential to analyze polynomial behavior at all points in the complex plane.

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 $v(z)^{-1}\prod_{k=1}^s(1-z\overline{a}_k)^{-1}$.