Nearest neighbor recurrence relations for multiple orthogonal polynomials

We show that multiple orthogonal polynomials for r measures (mu(1), ... , mu(r)) satisfy a system of linear recurrence relations only involving nearest neighbor multi-indices (n) over bar +/- (e) over bar (j), where (e) over bar (j) are the standard unit vectors. The recurrence coefficients are not arbitrary but satisfy a system of partial difference equations with boundary values given by the recurrence coefficients of the orthogonal polynomials with each of the measures mu(j). We show how the Christoffel-Darhoux formula for multiple orthogonal polynomials can be obtained easily using this information. We give explicit examples involving multiple Hermite, Charlier, Laguerre, and Jacobi polynomials. (C) 2011 Elsevier Inc. All rights reserved.