Multiple orthogonal polynomials on the unit circle. Normality and recurrence relations

摘要

Multiple orthogonal polynomials on the unit circle (MOPUC) were introduced by J. Mínguez and W. Van Assche for the first time in 2008. Some applications were given there and recurrence relations were obtained from a Riemann–Hilbert problem.This paper is a second contribution to this field. We first obtain a determinantal formula for MOPUC (multiple Heine’s formula) and we analyze the concept of normality, from a dynamical point of view and by presenting a first example: the combination of the Lebesgue and Rogers–Szegő measures. Secondly, we deduce recurrence relations for MOPUC without using Riemann–Hilbert analysis, only by considering orthogonality conditions. This new approach allows us to complete the recurrence relations in the situation when the origin is a zero of MOPUC, a case that was not considered before. As a consequence, we give an appropriate definition of multiple Verblunsky coefficients. A multiple version of the well known Szegő recurrence relation is also obtained. Here, the coefficients that appear in the recurrence satisfy certain partial difference equations that are used to present a recursive algorithm for the computation of MOPUC. A discussion on the Riemann–Hilbert approach that also includes the case when the origin is a zero of MOPUC is presented. Some conclusions and open questions are finally mentioned.