Multiple orthogonal polynomials of mixed type and non-intersecting Brownian motions

We present a generalization of multiple orthogonal polynomials of type I and type II, which we call multiple orthogonal polynomials of mixed type. Some basic properties are formulated, and a RiemannHilbert problem for the multiple orthogonal polynomials of mixed type is given. We derive a Christoffel-Darboux formula for these polynomials using the solution of the Riemann-Hilbert problem. The main motivation for studying these polynomials comes from a model of nonintersecting one-dimensional Brownian motions with a given number of starting points and endpoints. The correlation kernel for the positions of the Brownian paths at any intermediate time coincides with the Christoffel-Darboux kernel for the multiple orthogonal polynomials of mixed type with respect to Gaussian weights.

• Type: Journal
• Year: 2007
• Where: JAT
• Authors: Evi Daems, Arno B. J. Kuijlaars
• Keywords: JAT 2007, Mixed Type, Multiple Orthogonal Polynomials, One-dimensional Brownian Motions