Laurent Multiple Orthogonal Polynomials on the Unit Circle

Abstract: We introduce the notion of Laurent multiple orthogonal polynomials on the unit circle. These are the trigonometric polynomials that satisfy simultaneous orthogonality conditions with respect to several measures on the unit circle. This provides an alternative way to the approach of extending the usual theory of orthogonal polynomials on the unit circle to the multiple orthogonality case. We introduce Angelesco and AT systems, prove normality, and exhibit a connection to multiple orthogonal polynomials on the real line through the Szeg\H{o} mapping. We also present a generalized two-point Hermite-Pad\'e problem, whose solutions include multiple orthogonal polynomials of previous papers on the subject, as well as Laurent multiple orthogonal polynomials from the current paper, as special cases. Finally, we show that the Szeg\H{o} recurrence relations naturally extend to the solutions of this generalized Hermite-Pad\'e problem.