Szegő polynomials from hypergeometric functions

Abstract

Szegő polynomials with respect to the weight function ω ( θ ) = e η θ [ sin ⁡ ( θ / 2 ) ] 2 λ \omega (\theta ) = e^{\eta \theta } [\sin (\theta /2)]^{2\lambda } , where η , λ ∈ R \eta , \lambda \in \mathbb {R} and λ > − 1 / 2 \lambda > -1/2 are considered. Many of the basic relations associated with these polynomials are given explicitly. Two sequences of para-orthogonal polynomials with explicit relations are also given.