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An Introduction to Orthogonal Polynomials Read chapter Read first chapter

2020 | OriginalPaper | Chapter

Hypergeometric Multivariate Orthogonal Polynomials

Author : Iván Area

Published in: Orthogonal Polynomials

Publisher: Springer International Publishing

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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.

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previous chapter The Gamma Function
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Metadata
Title
Hypergeometric Multivariate Orthogonal Polynomials
Author
Iván Area
Copyright Year
2020
Book
Orthogonal Polynomials

Print ISBN: 978-3-030-36743-5

Electronic ISBN: 978-3-030-36744-2

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-36744-2_10

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