On a Series Representation for Carleman Orthogonal Polynomials

Let {pn(z)}n=0 be a sequence of complex polynomials (pn of degree n) that are orthonormal with respect to the area measure over the interior domain of an analytic Jordan curve. We prove that each pn of sufficiently large degree has a primitive that can be expanded in a series of functions recursively generated by a couple of integral transforms whose kernels are defined in terms of the degree n and the interior and exterior conformal maps associated with the curve. In particular, this series representation unifies and provides a new proof for two important known results: the classical theorem by Carleman establishing the strong asymptotic behavior of the polynomials pn in the exterior of the curve, and an integral representation that has played a key role in determining their behavior in the interior of the curve.