On a series representation for Carleman orthogonal polynomials

Abstract:

Let $\{p_n(z)\}_{n=0}^\infty$ be a sequence of complex polynomials ($p_n$ 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 $p_n$ 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 $p_n$ 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.