Nearest neighbor recurrence relations for multiple orthogonal polynomials

We show that multiple orthogonal polynomials for rr measures (μ1,…,μr)(μ1,…,μr) satisfy a system of linear recurrence relations only involving nearest neighbor multi-indices n→±e→j, where e→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 μjμj. We show how the Christoffel–Darboux formula for multiple orthogonal polynomials can be obtained easily using this information. We give explicit examples involving multiple Hermite, Charlier, Laguerre, and Jacobi polynomials.