2005/11/18 by E. Daems, A. B. J. Kuijlaars · 1 citation
Mathematics · #math.CA #math.PR
published as Journal of Approximation Theory 146 (2007), 91--114 · 22 pages, 4 figures
arxiv created 2005/11/18 · arxiv updated 2010/07/30
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 Riemann-Hilbert 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 non-intersecting 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.