Polynomials orthogonal on the semicircle

Abstract

Complex polynomials { π k }, π k ( z) = z k + ···, orthogonal with respect to the complex-valued inner product ( f, g) = ∝ 0 π f( e iθ ) g( e iθ ) dθ are studied. By direct calculation of moment determinants it is shown that these polynomials exist uniquely. The three-term recurrence relation satisfied by these polynomials is obtained explicitly as well as their relationship with Legendre polynomials. It is shown that the zeros of π n are all simple and are located in the interior of the upper unit half disc, distributed symmetrically with respect to the imaginary axis. They can be (and have been) computed as eigenvalues of a real nonsymmetric tridiagonal matrix. A linear second-order differential equation is obtained for π n ( z) which has regular singular points at z = 1, −1, ∞ (like Legendre's equation) and an additional regular singular point on the negative imaginary axis. Applications are discussed involving Gauss-Christoffel quadrature over the semicircle, numerical differentiation, and the computation of Cauchy principal value integrals.