The nearest neighbor recurrence coefficients for multiple orthogonal polynomials

Abstract

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 $\vec{n}\pm {\vec{e}}_j$, where ${\vec{e}}_j$ 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 measures ${\mu }_j$. 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.

22 pages

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