MULTIPLE ORTHOGONAL POLYNOMIALS ON THE SEMICIRCLE

In this paper multiple orthogonal polynomials on the semicircle, investigated by Milovanovie and Stanie in [Math. Balkanica (N. S.) 18 (2004), 373-387] (complex polynomials orthogonal with respect to the complex-valued inner products [f, g](m) = integral(pi)(0) f (e(i theta))g(e(i theta))w(m)(e(i theta)) d theta, for m = 1, 2, ... , r) are considered. These polynomials satisfy a linear recurrence relation of order r + 1. Under suitable assumption on the weight functions w(m), m = 1, 2, ... , r, we express multiple orthogonal polynomials on the semicircle in terms of the type II multiple orthogonal (real) polynomials with respect to the weight function w(m) (x), m = 1, 2, ... , r. Specially, we consider the case r = 2 and express coefficients of corresponding recurrence relations in terms of coefficients of recurrence relation for the type II multiple orthogonal (real) polynomials. In particular, we obtain these type of polynomials associated with Jacobi weight functions.