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

We present a generalization of multiple orthogonal polynomials of types I and 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.