Book review

It's certainly not necessary to introduce N.Ja. Vilenkin to those who have come into contact with special functions - his "Special functions and theory of group representations", which has lived to see its second edition recently, is a generally accepted textbook on this subject; and, together with the present co-author, Vilenkin also wrote the contribution about special functions and group representations into the Encyclopaedia of Mathematical Sciences (vol. 59, Non-commutative harmonic analysis II). The present book is probably (the third volume of) the climax of his work - an extensive (about 2000 pages) monograph about various special functions and their connections with Lie groups.

The topics treated in the present volume include: quantum groups, q-orthogonal polynomials and hypergeometric fw1ctions(136 pp. - the q-analogs of the classical special functions and polynomials), basics of semisimple Lie groups and homogeneous spaces (60 pp. - roots, Laplacians, invariant measures, ...), representations of Lie groups and their matrix elements (50 pp.), special functions of matrix argument (110 pp.), representations in the Gelfand-Tsetlin basis (95 pp.), and modular forms and related topics (theta functions, affine Lie algebras, the string function - 160 pp). There is a list of symbols, a (rather small) index, and an extensive bibliography (over three hundred items). The translation from Russian is good, and the typesetting is really excellent (TeX ?). There are also tables of contents of Volumes 1 and 2, which, though the reviewer has not yet been able to see the volumes themselves, gives some idea about the topics discussed in them. Volume 1 begins with some basics of harmonic analysis on Lie groups, and then deals with the special functions arising from representations of SO(2), SU(2), SL(2, C) and SL(2, R) (hypergeometric, Gamma, Bessel, Legendre, Jacobi, etc.). Volume 2 contains functions related to SO(n), U(n), the Heisenberg group, and discrete groups.

There are other books on special functions - too many of them, perhaps - but, especially regarding the connections with the Lie theory, none of them seems to treat the subject so comprehensively as this treatise. True, the price is stiff, and becomes even stiffer for all three volumes (the first one costs 155pounds, the second 183); but there is a lot in these books that can not readily be found elsewhere, and the authors belong to world's leading experts in the field. Given the ubiquity of special functions in mathematics, I think that at least libraries, if not individuals, should be persuaded to make the investment and buy.