Multiple Orthogonal Polynomials on the Semicircle and Corresponding Quadratures of Gaussian Type1

In this paper multiple orthogonal polynomials defined using orthogonality conditions spread out over r different measures are considered. We study multiple orthogonal polynomials on the real line, as well as on the semicircle (complex polynomials orthogonal with respect to the complex-valued inner products (f, g)k = ∫ π 0 f(e)g(e)wk(e ) dθ, for k = 1, 2, . . . , r). For r = 1, in the real case we have the ordinary orthogonal polynomials, and in complex case orthogonal polynomials on the semicircle, introduced by Gautschi and Milovanović [7]. Multiple orthogonal polynomials satisfy a linear recurrence relation of the order r + 1. This is a generalization of the second order linear recurrence relation for ordinary monic orthogonal polynomials (r = 1). Using the discretized Stieltjes-Gautschi procedure, we compute recurrence coefficients and also zeros of multiple orthogonal polynomials, as well as the weight coefficients for the corresponding quadrature formulas of Gaussian type. Some numerical examples are also included. AMS Subj. Classification: Primary 33C45, 42C05; Secondary 41A55, 65D30, 65D32.