Multiple orthogonal polynomials of mixed type and non-intersecting Brownian motions
Abstract
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.
Keywords
Cite
@article{arxiv.math/0511470,
title = {Multiple orthogonal polynomials of mixed type and non-intersecting Brownian motions},
author = {E. Daems and A. B. J. Kuijlaars},
journal= {arXiv preprint arXiv:math/0511470},
year = {2010}
}
Comments
22 pages, 4 figures