Encyclopedia of Mathematics and Its Applications. 81. Cambridge: Cambridge University Press. xv, 390 p. £ 55.00; $ 80.00 (2001).
This is the modern book on orthogonal polynomials of several variables, which are interesting both as objects of study and as tools used in multivariate analysis. The book presents the theory in elegant form and with modern concepts and notation. The book contains 9 chapters. Chapter 1 is a summary of the key one variable methods and definitions: gamma and beta functions, the classical orthogonal polynomials and their structure constants, hypergeometric and Lauricella series. The multi-variable analysis begins in Chapter 2 with some examples of orthogonal polynomials of several variables and spherical harmonics. Specific two variables examples such as Jacobi polynomials on various domains and disc polynomials are considered.
Chapter 3 presents general properties of orthogonal polynomials of 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. The authors adopt the point of view which make the basis of results independent. This allows to derive a proper analogy of the three term recurrence relations for orthogonal polynomials of several variables, to define block Jacobi matrices and to study them as self-adjoint operators, and to investigate common zeros of orthogonal polynomials of several variables. Chapter 4 treats systematically (in a self-contained way) Coxeter groups. This exposition is given in a style suitable for an analyst (knowledge of the representation theory is not necessary). The chapter goes on to introduce differential-difference operators, the intertwining operator, and the analogue of the exponential function, and concludes with the construction of invariant differential operators.
Chapter 5 presents $h$-harmonics (the analogue of harmonic homogeneous polynomials) associated with reflection groups.There are some examples for specific reflection groups as well as the application to proving the isometric properties of the generalized Fourier transform. This transform uses the analogue of the exponential function. It contains the classical Hankel transform as a special case. Chapter 6 studies orthogonal polynomials that generalize the classical orthogonal polynomials in the sense that they satisfy a second order differential-difference equation. They are orthogonal with respect to weight functions that contain a number of parameters, and they reduce to the classical orthogonal polynomials when some of the parameters go to zero. Explicit formulas for reproducing kernels are derived for some families of orthogonal polynomials.
The basics of Fourier orthogonal expansion are discussed in Chapter 7. Various convergence results of the orthogonal expansion are derived. The authors prove results on the summability of the Cesàro means of the orthogonal expansions for $h$-harmonics and the generalized classical orthogonal polynomials. In Chapter 8 the nonsymmetric Jack polynomials appear. This chapter contains all necessary details for their derivation, formulas for norms, hook length products, and computation of the structure constants. There is a proof of the Macdonald-Mehta-Selberg integral formula.
Chapter 9 shows how to use the nonsymmetric Jack polynomials to produce bases associated with the octahedral groups. This chapter discusses how these polynomials and related operators are used to solve the Schrödinger equations of the Calogero--Sutherland systems (these are exactly solvable models of quantum mechanics involving identical particles in a one-dimensional space). The book can be used as an introduction to the subject and as a reference book. The book will be useful for research mathematicians and applied scientists, including applied mathematicians and physicists.