AIMSVSW 2018: Orthogonal Polynomials pp 165-193

# Hypergeometric Multivariate Orthogonal Polynomials

Conference paper
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)

## Abstract

In this lecture a comparison between univariate and multivariate orthogonal polynomials is presented. The first step is to review classical univariate orthogonal polynomials, including classical continuous, classical discrete, their q-analogues and also classical orthogonal polynomials on nonuniform lattices. In all these cases, the orthogonal polynomials are solution of a second-order differential, difference, q-difference, or divided-difference equation of hypergeometric type. Next, a review multivariate orthogonal polynomials is presented. In the approach we have considered, the main tool is the partial differential, difference, q-difference or divided-difference equation of hypergeometric type the polynomial sequences satisfy. From these equations satisfied, the equation satisfied by any derivative (difference, q-difference or divided-difference) of the polynomials is obtained. A big difference appears for nonuniform lattices, where bivariate Racah and for bivariate q-Racah polynomials satisfy a fourth-order divided-difference equation of hypergeometric type. From this analysis, we propose a definition of multivariate classical orthogonal polynomials. Finally, some open problems are stated.

## Keywords

Orthogonal polynomials Hypergeometric equation

## Mathematics Subject Classification (2000)

Primary 33C45 33C50; Secondary 39A13 39A14

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