Multiple orthogonal polynomials on the semicircle and applications

In this paper two types of multiple orthogonal polynomials on the semicircle with respect to a set of r different weight functions are defined. Such polynomials are generalizations of polynomials orthogonal on the semicircle with respect to a complex-valued inner product [f,g]=∫0πfeiθgeiθweiθdθ. The existence and uniqueness of introduced multiple orthogonal polynomials for certain classes of weight functions are proved. Some properties of multiple orthogonal polynomials on the semicircle including certain recurrence relations of order r+1 are presented. Finally, an application in numerical integration is given.