A New Class of Orthogonal Polynomials on the Unit Circle

Even though the theory of orthogonal polynomials on the unit circle, also known as the theory of Szegő polynomials, is very extensive, it is less known than the theory of orthogonal polynomials on the real line. One reason for this may be that “beautiful” examples on the theory of Szegő polynomials are scarce. This is in contrast to the wonderful examples of Jacobi, Laguerrer and Hermite polynomials with regards to the theory of orthogonal polynomials on the real line. The Jacobi polynomials are orthogonal polynomials with respect to the class of weight functions (1 − x)α(1 + x)β on (−1, 1), where the parameters α, β are such that α > −1 and β > −1. The nth degree Jacobi polynomials is also the Hypergeometric polynomial given by 2F1(−n, n+ α+ β;α+ 1; (1− x)/2). The object of this exposition is to present the new example of Szegő polynomials with respect to the class of weight functions ω(θ) = e−ηθ[sin(θ/2)]2λ, where the parameters η, λ are such that η, λ ∈ R and λ > −1/2. It is shown that the associated nth degree Szegő polynomial is the Hypergeometric polynomial 2F1(−n, b+ 1; b+ b̄+ 1; 1− z), where λ = Re b and η = Im b. Given a positive measure μ(z) = μ(eiθ) on the unit circle C = {z = eiθ : 0 ≤ θ ≤ 2π}, the associated sequence of “monic” Szegő polynomials {Φn}n=0 can be defined by ∫