Orthogonal polynomials in weighted Bergman spaces

Abstract

Let w be a weight on the unit disk D having the form w(z)=|v(z)|2∏k=1sz−ak1−za¯kmk,mk>−2,|ak|<1,where v is analytic and free of zeros in D¯, and let (pn)n=0∞ be the sequence of polynomials (pn of degree n) orthonormal over D with respect to w. We give an integral representation for pn from which it is in principle possible to derive its asymptotic behavior as n→∞ 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∏k=1s(1−za¯k)−1.