Szeg\H{o} Recurrence for Multiple Orthogonal Polynomials on the Unit Circle

TL;DR

This paper extends Szegő recurrence relations to multiple orthogonal polynomials on the unit circle, introducing new coefficients and formulas that generalize classical results to a multi-measure setting.

Contribution

It generalizes Szegő recurrence relations and identifies Verblunsky coefficient analogues for multiple orthogonal polynomials on the unit circle.

Findings

Generalized Szegő recurrence relations for multiple orthogonal polynomials.

Identified analogues of Verblunsky coefficients in this setting.

Proved the Christoffel–Darboux formula for these polynomials.

Abstract

We investigate polynomials that satisfy simultaneous orthogonality conditions with respect to several measures on the unit circle. We generalize the direct and inverse Szeg\H{o} recurrence relations, identify the analogues of the Verblunsky coefficients, and prove the Christoffel$\unicode{x2013}$Darboux formula. These results stand directly in analogue with the nearest neighbour recurrence relations from the real line counterpart.