  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.
