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

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.