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.

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 Fériet [1926] and Erdélyi et al. [1953]. Twenty-five years have gone by since Koornwinder's survey article [1975]. A number of individuals who need techniques from this topic have approached us and suggested (even asked) that we write a book accessible to a general mathematical audience.

It is our goal to present the developments of very recent research to a readership trained in classical analysis. We include applied mathematicians and physicists, and even chemists and mathematical biologists, in this category.

While there is some material about the general theory, the emphasis is on classical types, by which we mean families of polynomials whose weight functions are supported on standard domains such as the simplex and the ball, or Gaussian types, which satisfy differential–difference equations and for which fairly explicit formulae exist. The term “difference” refers to operators associated with reflections in hyperplanes. The most desirable situation occurs when there is a set of commuting self-adjoint operators whose simultaneous eigenfunctions form an orthogonal basis of polynomials. As will be seen, this is still an open area of research for some families.

In this chapter we consider orthogonal polynomials with respect to a weight function defined on the unit sphere, the structure of which is not covered by the discussion in the previous chapter. Indeed, if dμis a measure supported on the unit sphere of ℝd then the linear functional ℒ(f) = ∫ f dε is not positive definite in the space of polynomials, as ∫(1 − ∥x∥2)2 dµ = 0. It is positive definite in the space of polynomials restricted to the unit sphere, which is the space in which these orthogonal polynomials are defined.

We consider orthogonal polynomials with respect to the surface measure on the sphere first; these are the spherical harmonics. Our treatment will be brief, since most results and proofs will be given in a more general setting in Chapter 7. The general structure of orthogonal polynomials on the sphere will be derived from the close connection between the orthogonal structures on the sphere and on the unit ball. This connection goes both ways and can be used to study classical orthogonal polynomials on the unit ball. We will also discuss a connection between the orthogonal structures on the unit sphere and on the simplex.

Spherical Harmonics

The Fourier analysis of continuous functions on the unit sphere Sd−1 := {x:∥x∥ = 1} in ℝd is performed by means of spherical harmonics, which are the restrictions of homogeneous harmonic polynomials to the sphere. In this section we present a concise overview of the theory and a construction of an orthogonal basis by means of Gegenbauer polynomials. Further results can be deduced as special cases of theorems in Chapter 7, by taking the weight function there as 1.

In this second edition, several major changes have been made to the structure of the book. A new chapter on orthogonal polynomials in two variables has been added to provide a more convenient source of information for readers concerned with this topic. The chapter collects results previously scattered in the book, specializing results in several variables to two variables whenever necessary, and incorporates further results not covered in the first edition. We have also added a new chapter on orthogonal polynomials on the unit sphere, which consolidates relevant results in the first edition and adds further results on the topic. Since the publication of the first edition in 2001, considerable progress has been made in this research area. We have incorporated several new developments, updated the references and, accordingly, edited the notes at the ends of relevant chapters. In particular, Chapter 5, “Examples of Orthogonal Polynomials in Several Variables”, has been completely rewritten and substantially expanded. New materials have also been added to several other chapters. An index of symbols is given at the end of the book.

Another change worth mentioning is that orthogonal polynomials have been renormalized. Some families of orthogonal polynomials in several variables have expressions in terms of classical orthogonal polynomials in one variable. To provide neater expressions without constants in square roots they are now given in the form of orthogonal rather than orthonormal polynomials as in the first edition. The L2 norms have been recomputed accordingly.

In this chapter we consider analysis associated with symmetric groups. The differential–difference operators for these groups, called type A in Weyl group nomenclature, are crucial in this theory. The techniques tend to be algebraic, relying on methods from combinatorics and linear algebra. Nevertheless the chapter culminates in explicit evaluations of norm formulae and integrals of the Macdonald–Mehta–Selberg type. These integrals involve the weight function Π1≤i<j≤d|xi − xj|2κ on the torus and the weight function on ℝd equipped with the Gaussian measure. The fundamental objects are a commuting set of self-adjoint operators and the associated eigenfunction decomposition. The simultaneous eigenfunctions are certain homogeneous polynomials, called nonsymmetric Jack polynomials. The Jack polynomials are a family of parameterized symmetric polynomials, which have been studied mostly in combinatorial settings.

The fact that the symmetric group is generated by transpositions of adjacent entries will frequently be used in proofs; for example, it suffices to prove invariance under adjacent transpositions to show group invariance. Two bases of polynomials will be used, not only the usual monomial basis but also the p-basis; these are polynomials, defined by a generating function, which have convenient transformation formulae for the differential–difference operators. Also, they provide expressions for the nonsymmetric Jack polynomials which are independent of the number of trailing zeros of the label α ∈ ℕd0.

In this chapter we present the general properties of orthogonal polynomials in several variables, that is, those properties that hold for orthogonal polynomials associated with weight functions that satisfy some mild conditions but are not any more specific than that.

This direction of study started with the classical work of Jackson [1936] on orthogonal polynomials in two variables. It was realized even then that the proper definition of orthogonality is in terms of polynomials of lower degree and that orthogonal bases are not unique. Most subsequent early work was focused on understanding the structure and theory in two variables. In Erdélyi et al. [1953], which documents the work up to 1950, one finds little reference to the general properties of orthogonal polynomials in more than two variables, other than (Vol. II, p. 265): “There does not seem to be an extensive general theory of orthogonal polynomials in several variables.” It was remarked there that the difficulty lies in the fact that there is no unique orthogonal system, owing to the many possible orderings of multiple sequences. And it was also pointed out that since a particular ordering usually destroys the symmetry, it is often preferable to construct biorthogonal systems. Krall and Sheffer [1967] studied and classified two-dimensional analogues of classical orthogonal polynomials as solutions of partial differential equations of the second order.