On polynomials orthogonal on the semicircle and applications

AbstractPolynomials{πn} orthogonal on the semicircle Γ={zϵC:z=eiθ,0⩽θ⩽π} with respect to the inner product (f,g)=ʃγf(z)g(z)w(z)(iz)−1dz, where z ↦ w(z) is a complex weight function, have been introduced in 1986–1987 by Gautschi, Landau and the author. In this paper we introduce the functions of the second kind, as well as the corresponding associated polynomials, and prove some recurrence relations. For Gauss-Gegenbauer quadrature formulae over the semicircle, applied to analytic functions, we develop error bounds from contour integral representations of the remainder term and give some numerical results