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arXiv · 1104.3778

The nearest neighbor recurrence coefficients for multiple orthogonal polynomials

Walter Van Assche
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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.

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Walter Van Assche. 2011-04-19. The nearest neighbor recurrence coefficients for multiple orthogonal polynomials. https://doi.org/10.1016/j.jat.2011.05.003

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