The study of orthogonal polynomials of several variables goes back at least as far as Hermite. There have been only a few books on the subject since: Appell and de Feriet and Erddlyi et al. Twenty-five years have gone by since Koornwinder's survey article . A number of individuals who need techniques from this topic have approched us and suggested (even asked) that we write a book accessible to a general mathematical audience.

Preface 

1 Background

 1.1 The Gamma and Beta Functions

 1.2 Hypergeometric Series

 1.3 Orthogonal Polynomials of One Variable

1.3.1 General properties

1.3.2 Three term recurrence

 1.4 Classical Orthogonal Polynomials

1.4.1 Hermite polynomials

1.4.2 Laguerre polynomials

1.4.3 Gegenbauer polynomials

1.4.4 Jacobi polynomials

 1.5 Modified Classical Polynomials

1.5.1 Generalized Hermite polynomials

1.5.2 Generalized Gegenbauer polynomials

1.5.3 A limiting relation

 1.6 Notes

2 Examples of Orthogonal Polynomials in Several Variables

 2.1 Notation and Preliminary

 2.2 Spherical Harmonics

 2.3 Classical Orthogonal Polynomials

2.3.1 Multiple Jacobi polynomials on the cube

2.3.2 Classical orthogonal polynomials on the unit ball

2.3.3 Classical orthogonal polynomials on the simplex

2.3.4 Multiple Hermite polynomials on Rd

2.3.5 Multiple Laguerre polynomials on

 2.4 Other Examples of Orthogonal Polynomials

2.4.1 Two general families of orthogonal polynomials

2.4.2 A method for generating orthogonal polynomials of two variables

2.4.3 Disc polynomials

 2.5 Van der Corput-Schaake Inequality

 2.6 Notes

3 General Properties of Orthogonal Polynomials in Several Variables

 3.1 Moment Functionals and Orthogonal Polynomials in Several Variables

3.1.1 Definition of orthogonal polynomials

3.1.2 Orthogonal polynomials and moment matrices

3.1.3 The moment problem

 3.2  The Three Term Relation

3.2.1 Definition and basic properties

3.2.2 Favard's theorem

3.2.3 Centrally symmetric integrals

3.2.4 Examples

 3.3 Jacobi Matrices and Commuting Operators

 3.4 Further Properties of the Three Term Relation

3.4.1 Recurrence formula

3.4.2 General solutions of the three-term relation

 3.5 Reproducing Kernels and Fourier Orthogonal Series

3.5.1 Reproducing kernels

3.5.2 Fourier orthogonal series

 3.6 Common Zeros of Orthogonal Polynomials in Several Variables

 3.7 Gaussian Cubature Formulae

3.7.1 Characterization of Gaussian cubature formulae

3.7.2 Examples of Gaussian cubature formulae

 3.8 Orthogonal Polynomials on the Unit Sphere

3.8.1 Orthogonal structures on Sd and on Bd

3.8.2 Orthogonal structure on Bd and on Sd+m

 3.9 Notes

4 Root Systems and Coxeter groups

 4.1 Introduction and Overview

 4.2 Root Systems

4.2.1 Type Ad-1

4.2.2 Type Bd

4.2.3 Type I2(m)

4.2.4 Type Dd

4.2.5 Type/-/3

4.2.6 Type F4

4.2.7 Other types

4.2.8 Miscellaneous results

 4.3  Invariant Polynomials

4.3.1 Type Ad-1 invariants

4.3.2 Type Bd invariants

4.3.3 Type Dd invariants

4.3.4 Type/2 (m) invariants

4.3.5 Type Ha invariants

4.3.6 Type F4 invariants

 4.4 Differential-Difference Operators

 4.5 The Intertwining Operator

 4.6 The x-Analog of the Exponential

 4.7 Invariant Differential Operators

 4.8 Notes

5 Spherical Harmonics Associated with Reflection Groups

 5.1 h-Harmonic Polynomials

 5.2 Inner Products on Polynomials

 5.3 Reproducing Kernels and the Poisson Kernel

 5.4 Integration of the Intertwining Operator

 5.5 Example: Abelian Group Zd

 5.6 Example: Dihedral Groups

5.6.1 An orthonormal basis of ■

5.6.2 Canchy and Poisson kernels

 5.7 The Fourier Transform

 5.8 Notes

6 Classical and Generalized Classical Orthogonal Polynomials

 6.1 Generalized Classical Orthogonal Polynomials on the Ball

  6.1.1 Definition and differential-difference equations

  6.1.2 Bases, reproducing kernels, and the Funk-Hecke formula

 6.2 Orthogonal Polynomials on the Simplex

  6.2.1 General weight functions on Td

  6.2.2 Generalized classical orthogonal polynomials

 6.3 Generalized Hermite Polynomials

 6.4 Generalized Laguerre Polynomials

 6.5 Notes

7 Summability of Orthogonal Expansions

 7.1 General Results on Orthogonal Expansions

7.1.1 Uniform convergence of partial sums

7.1.2 Ceshro means of the orthogonal expansion

 7.2 Orthogonal Expansion on the Sphere

 7.3 Orthogonal Expansion on the Ball

 7.4 Orthogonal Expansions on the Simplex

 7.5 Orthogonal Expansion of Laguerre and Hermite Polynomials

 7.6 Multiple Jacobi Expansion

 7.7 Notes

8 Orthogonal Polynomials Associated with Symmetric Groups

 8.1 Introduction

 8.2 Partitions，Compositions and Orderings

 8.3 Commuting Self-Adjoint Operators

 8.4 The DUal Polynomial Basis

 8.5 Sd Invariant Subspaces

 8.6 Degree Changing Recurrences

 8.7 Norm Formulae

8.7.1 Hook length products and the pairing norm

8.7.2 The biorthogonal type norm

8.7.3 The torus inner product

8.7.4 Normalizing constants

 8.8 Symmetric Functions and Jack Polynomials

 8.9 Miscellaneous Topics

 8.10 Notes

9 Orthogonal Polynomials Associated with Octahedral Groups and Applications

 9.1 Introduction

 9.2 Operators of Type B

 9.3 Polynomial Eigenfunctions of Type B

 9.4 Generalized Binomial Coemcients

 9.5 Hermite Polynomials of Type B

 9.6 Calogero-Sutherland Systems

9.6.1 The simple harmonic oscillator

9.6.2 Root systems and the Laplacian

9.6.3 Type A models on the line

9.6.4 Type A models on the circle

9.6.5 Type B models on the line

 9.7 Notes

Bibliography

Author index

Symbol index

Subject index