The nearest neighbor recurrence coefficients for multiple orthogonal polynomials

TL;DR

This paper establishes that multiple orthogonal polynomials satisfy nearest neighbor recurrence relations with coefficients obeying partial difference equations, providing explicit examples and a new derivation of the Christoffel-Darboux formula.

Contribution

It introduces a systematic way to derive recurrence relations and coefficients for multiple orthogonal polynomials using partial difference equations and boundary conditions.

Findings

Recurrence relations involve only nearest neighbor multi-indices.

Recurrence coefficients satisfy partial difference equations.

Explicit examples for classical multiple orthogonal polynomials are provided.

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.