The nearest neighbor recurrence coefficients for multiple orthogonal polynomials

Details The nearest neighbor recurrence coefficients for multiple orthogonal polynomials The nearest neighbor recurrence coefficients for multiple orthogonal polynomials

2011-04-19

arxiv.org/abs/1104.3778v1

J. Approx. Theory 163 (2011), 1427-1448

Mathematics

Classical Analysis and ODEs

22 pages

Scientific paper

10.1016/j.jat.2011.05.003

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.

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