Mathematics and Its Applications. Soviet Series. 72. Dordrecht etc.: Kluwer Academic Publishers. xxiii, 608 p. (1991).

This is volume 1 of a three volume updating of the first (and recently deceased) author’s classic book “Special Functions and the Theory of Group Representations” (1965;

Zbl 0144.380) which became available in English translation in 1968. The point of view is, essentially, 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. If the basis is orthonormal then the group homomorphism property of the matrix elements leads to addition theorems for the associated special functions. Corresponding to a continuum basis, these addition theorems can be interpreted as integral transforms. Taking advantage of recent advances in the field, the authors also consider Clebsch-Gordan coefficients and Racah coefficients for group representations of the simplest Lie group as special functions of a discrete argument. Then the unitarity conditions for matrix elements, Clebsch-Gordan and Racah coefficients can (typically) be interpreted as orthogonality relations for families of polynomials of a discrete variable. In comparison with the two volumes to follow, this book contains relatively few new results. It is a considerable expansion of the first eight of the eleven chapters in the 1968 original, the pace is more leisurely and less focussed. (For example, the first 108 pages are devoted to a review and summary of the basic facts about groups, algebras and their representations; only following this do the most elementary special functions appear.) Then in successive chapters the authors treat commutative groups (exponential functions, Fourier transforms), the $ax+b$ group (gamma functions), Euclidean and pseudo-Euclidean groups in the plane (cylindrical functions), a group of third order triangular matrices (confluent hypergeometric functions), $SU(2)$ and $SU(1,1)$ (Gaussian hypergeometric functions) and, finally, Clebsch-Gordan and Racah coefficients for $SU(2)$ and $SU(1,1)$ (generalized hypergeometric functions).
The principal new material, compared to the original, concerns more results on matrix elements in continuum bases and, most significantly, a group theoretic treatment of families of polynomials orthogonal with respect to a discrete variable (including the Wilson polynomials through their relation to Racah coefficients.
Volume 2 will take up special functions of several arguments, associated to groups in $n$ dimensions: $SO(n)$, $U(n)$, $ISO(n)$ and the Heisenberg groups.
Volume 3 is concerned with quantum groups, general semisimple Lie groups, and special functions of a matrix argument, as well as modular forms, theta functions and representations of affine Lie algebras. The complete opus, providing easy access to a large share of the known connections between groups and special functions, should prove extremely valuable to the scientific community and a fitting monument to N. Ya. Vilenkin.