Szeg\H{o} Mapping and Hermite--Pad\'{e} Polynomials for Multiple Orthogonality on the Unit Circle

TL;DR

This paper explores generalized Laurent multiple orthogonal polynomials on the unit circle, characterizing them via Hermite-Padé approximation, recurrence relations, and linking them to polynomials on the real line.

Contribution

It introduces a new framework connecting multiple orthogonality on the unit circle with Hermite-Padé approximation and real line polynomials, extending classical mappings.

Findings

Derived Szegő-type recurrence relations

Established compatibility conditions for recurrence coefficients

Connected unit circle orthogonality to real line polynomials

Abstract

We investigate generalized Laurent multiple orthogonal polynomials on the unit circle satisfying simultaneous orthogonality conditions with respect to $r$ probability measures or linear functionals on the unit circle. We show that these polynomials can be characterized as solutions of a general two-point Hermite--Pad\'e approximation problem. We derive Szeg\H{o}-type recurrence relations, establish compatibility conditions for the associated recurrence coefficients, and obtain Christoffel--Darboux formulas as well as Heine-type determinantal representations. Furthermore, by extending the Szeg\H{o} mapping and the Geronimus relations, we relate these Laurent multiple orthogonal polynomials to multiple orthogonal polynomials on the real line, thereby making explicit the connection between multiple orthogonality on the unit circle and on the real line.