Szegő Recurrence for Multiple Orthogonal Polynomials on the Unit Circle

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.