Average characteristic polynomials for multiple orthogonal polynomial ensembles

Multiple orthogonal polynomials (MOP) are a non-definite version of matrix orthogonal polynomials. They are described by a Riemann–Hilbert matrix YY consisting of four blocks Y1,1Y1,1, Y1,2Y1,2, Y2,1Y2,1 and Y2,2Y2,2. In this paper, we show that detY1,1detY1,1 (detY2,2detY2,2) equals the average characteristic polynomial (average inverse characteristic polynomial, respectively) over the probabilistic ensemble that is associated to the MOP. In this way we generalize the classical results for orthogonal polynomials, and also some recent results for MOP of type I and type II. We then extend our results to arbitrary products and ratios of characteristic polynomials. In the latter case an important role is played by a matrix-valued version of the Christoffel–Darboux kernel. Our proofs use determinantal identities involving Schur complements, and adaptations of the classical results by Heine, Christoffel and Uvarov.