Nearest neighbor recurrence relations for multiple orthogonal polynomials

We show that multiple orthogonal polynomials for r measures ( μ 1 , … , μ r ) 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 p...

Full description

Saved in:
Bibliographic Details
Published in:Journal of approximation theory 2011-10, Vol.163 (10), p.1427-1448
Main Author: Van Assche, Walter
Format: Article
Language:English
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We show that multiple orthogonal polynomials for r measures ( μ 1 , … , μ r ) 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 μ 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.
ISSN:0021-9045
1096-0430
DOI:10.1016/j.jat.2011.05.003