This is a follow up to the authors’ three volume updating of N. Vilenkin’s classic book “Special Functions and the Theory of Group Representations” (1965; Zbl 0144.380). In the earlier books (Vol. I, 1991; Zbl 0742.22001; Vol. II, 1993; Zbl 0809.22001; Vol. III, 1992; Zbl 0778.22001) the point of view is that special functions are spherical functions on groups, i.e., they are matrix elements of irreducible group representations with respect to appropriate bases and in appropriate group coordinates. This volume contains more recent results relating special functions and group representations, but which do not fit into the spherical function framework.
There are six chapters. The first is devoted to Coxeter groups, \(h\)- harmonic polynomials, and \(h\)-Hankel transforms. Coxeter groups are finite subgroups of the orthogonal groups \(O(n)\) that are generated by reflections. Corresponding to the natural action of a Coxeter group \(G\) in \(n\)-dimensional Euclidean space \(E_n\) and a function \(h({\mathbf x})\) on \(E_n\), Dunkl has defined a family of differential difference operators \(T_i\), \(i = 1,\dots, n\), that are analogies of the differential operators \(\partial_{x_i}\). The \(h\)-Laplacian is the differential difference operator defined by \(\Delta_h = T^2_1 + \cdots + T^2_n\); it has the property that it commutes with the action of \(G\). The authors study \(h\)-harmonic polynomials, i.e., polynomial solutions \(P\) in \(E_n\) of the equation \(\Delta_h P = 0\). There is a beautiful analogy between the theory of \(h\)-harmonic polynomials and the representation theory of \(G\) on one hand, and ordinary harmonic polynomials and the representation theory of \(O(n)\) on the other. Similarly the study of eigenfunctions of \(\Delta_h\) leads to analogies of Bessel functions that arise in the computation of eigenfunctions of the ordinary Laplacian.
Chapter 2 concerns symmetric polynomials and symmetric functions. These are functions in many variables, invariant under the natural action of the symmetric group. Among the special functions studied are Schur functions (associated with the representation theory of \(S_n\) and \(G \ell (n,C)\)), Hall-Littlewood polynomials, Jack polynomials (a more general one-parameter family of functions) and Macdonald polynomials. A very useful technique in this study is the introduction of various inner products on the space of symmetric polynomials with respect to which these special functions are orthogonal or biorthogonal. Both Jack and Macdonald polynomials are characterized as simultaneous eigenfunctions of families of \(n\) commuting differential operators. As pointed out in earlier volumes of this series, \(S_n\) is the Weyl group of the root system of the complex simple Lie algebra \(G \ell (n,C)\). In the remainder of this chapter the authors study Macdonald polynomials associated with the root systems of the other complex simple Lie algebras.
Chapter 3 is concerned with hypergeometric functions related to Jack polynomials. These are multivariable symmetric functions that are built from standard multivariable hypergeometric power series, essentially by replacing the powers of the variables \(x_i\) by Jack polynomials. There are many analogies to ordinary \(r\)-variable hypergeometric functions, including a characterization of these functions as simultaneous solutions of \(r\) second order partial differential equations. As in the preceding chapter, the authors further generalize their results by considering hypergeometric functions associated to root systems of complex simple Lie algebras.
Chapter 4 is devoted to the general study of Clebsch-Gordan coefficients and, in particular, Racah coefficients of finite dimensional group representations. The point of view is that the unitarity conditions and other special properties of these coefficients can be viewed as orthogonality relations for systems of special functions of many discrete variables. In Chapter 5 these results are specialized to the unitary groups \(U(n)\), where also multivariable hypergeometric series adapted to \(U(n)\) symmetry (and related to the Racah coefficients) are introduced. Much of the work in this chapter is due to Holman, Biedenharn, Louck, Gustafson and Milne.
The final chapter is devoted to Gel’fand hypergeometric functions. Basically, these are families of multivariable hypergeometric series in \(n\) variables whose \(m\) parameters can be considered as lying on a lattice. First order differential recurrence relations obeyed by the family are interpreted as translations on the lattice of parameters. By a suitable introduction of new variables \(u_i\), \(i = 1,\dots, n+m\), a subset of \(m + n\) of these recurrence relations can be considered simply as the application of the differential operator \(T_i = \partial_{u_i}\). From the structure of the lattice, one can show that the hypergeometric functions \(\Phi\) satisfy systems of partial differential equations of the form \((T_{i_1} \dots T_{i_s} - T_{i_{s+1}} \dots T_{i_t}) \Phi = 0\). Indeed these functions are characterized as solutions of the system which satisfy certain homogeneity relations in the variables. \(q\)-analogs of some of the results are obtained. There is also consideration given to hypergeometric functions on Grassmannians, and relations to Radon transforms.
There is a very extensive bibliography. This volume should prove quite useful as a source of detailed information on the topics that it covers. In common with the first three volumes, and in contrast with Vilenkin’s original book, it is not very stimulating to read. It is clear and reasonably complete but there are few digressions to explain what is important and what is not. The entire opus with its assimilation of a vast amount of research material is a very impressive achievement.