PDEA Seminar: Recurrence Relations for Multiple Orthogonal Polynomials

Abstract: The theory of orthogonal polynomials is an old and well-developed theory, for measures on the real line as well as measures on the unit circle. A generalization of this theory, called multiple orthogonal polynomials, has developed a lot in the last few decades, motivated through Hermite-Padé rational approximations of analytic functions. So far, mainly real line measures have been considered. In recent work together with Rostyslav Kozhan we extend some of this theory to unit circle measures. In particular, we present recurrence relations that generalize the Szegő recurrence from orthogonal polynomials on the unit circle, while at the same time being a perfect analogue of the popular nearest-neighbour recurrence relations from multiple orthogonal polynomials of the real line (by Van Assche, 2011). The generated recurrence coefficients will satisfy a set of partial difference equations, nearly identical to the real line equivalent result, and this also implies a similar Christoffel-Darboux formula. In this talk, I will mention some of the most famous results from both orthogonal polynomials and multiple orthogonal polynomials, as well as discuss how our recent research on multiple unit circle measures relates to these results. 

This is a seminar in the PDEs and Applications seminar series