Book contents
- Frontmatter
- Contents
- Preface to the Second Edition
- Preface to the First Edition
- 1 Background
- 2 Orthogonal Polynomials in Two Variables
- 3 General Properties of Orthogonal Polynomials in Several Variables
- 4 Orthogonal Polynomials on the Unit Sphere
- 5 Examples of Orthogonal Polynomials in Several Variables
- 6 Root Systems and Coxeter Groups
- 7 Spherical Harmonics Associated with Reflection Groups
- 8 Generalized Classical Orthogonal Polynomials
- 9 Summability of Orthogonal Expansions
- 10 Orthogonal Polynomials Associated with Symmetric Groups
- 11 Orthogonal Polynomials Associated with Octahedral Groups, and Applications
- References
- Author Index
- Symbol Index
- Subject Index
11 - Orthogonal Polynomials Associated with Octahedral Groups, and Applications
Published online by Cambridge University Press: 05 August 2014
- Frontmatter
- Contents
- Preface to the Second Edition
- Preface to the First Edition
- 1 Background
- 2 Orthogonal Polynomials in Two Variables
- 3 General Properties of Orthogonal Polynomials in Several Variables
- 4 Orthogonal Polynomials on the Unit Sphere
- 5 Examples of Orthogonal Polynomials in Several Variables
- 6 Root Systems and Coxeter Groups
- 7 Spherical Harmonics Associated with Reflection Groups
- 8 Generalized Classical Orthogonal Polynomials
- 9 Summability of Orthogonal Expansions
- 10 Orthogonal Polynomials Associated with Symmetric Groups
- 11 Orthogonal Polynomials Associated with Octahedral Groups, and Applications
- References
- Author Index
- Symbol Index
- Subject Index
Summary
Introduction
The adjoining of sign changes to the symmetric group produces the hyperoctahedral group. Many techniques and results from the previous chapter can be adapted to this group by considering only functions that are even in each variable. A second parameter κ′ is associated with the conjugacy class of sign changes. The main part of the chapter begins with a description of the differential–difference operators for these groups and their effect on polynomials of arbitrary parity (odd in some variables, even in the others). As in the type-A case there is a fundamental set of first-order commuting self-adjoint operators, and their eigenfunctions are expressed in terms of nonsymmetric Jack polynomials. The normalizing constant for the Hermite polynomials, that is, the Macdonald–Mehta–Selberg integral, is computed by the use of a recurrence relation and analytic-function techniques. There is a generalization of binomial coefficients for the nonsymmetric Jack polynomials which can be used for the calculation of the Hermite polynomials. Although no closed form is as yet available for these coefficients, we present an algorithmic scheme for obtaining specific desired values (by symbolic computation). Calogero and Sutherland were the first to study nontrivial examples of many-body quantum models and to show their complete integrability.
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- Chapter
- Information
- Orthogonal Polynomials of Several Variables , pp. 364 - 395Publisher: Cambridge University PressPrint publication year: 2014