On a series representation for Carleman orthogonal polynomials

Abstract

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