
Author(s) 
Vilenkin, N.Ja. Klimyk, A.U. 

Title  Representation of Lie Groups and Special Functions, Vol. 2: Class I Representations, Special Functions, and Integral Transforms 
Publisher  Kluwer Academic Publishers 
Year of publication  1993 
Reviewed by  VALERII V.TROFIMOV 
Numerous and mysterious connections between the vast numbers of the special functions of the mathematical physics are existed. The representation theory of groups gives the causes for these relations. The second volume contains the special functions concerned with the representations of the orthogonal groups, unitary groups, Heisenberg groups and discrete groups. Same nonclassical topics are presented here. The infinite dimensional Laplace operator and Hermite polynomials are discussed. Groups of linear transformations over finite fields and analogs of special functions in this case are studied as well. To construct the spesial functions we have to choose the basis of the representation space. General methods of the tracing the required basis are given.
One of the authors of the present volume is creator of this direction of the mathematical physics. I think, this threevolume encyclopedia supplies all that is known in this area. The rough contents of the second volume are as follows: 9. Special Functions Connected with SO(n) and with Related Groups, 10. Representation of Groups, Related to SO(n1), in NonCanonical Bases, Special Functions, and Integral Transforms, 11. Special Functions Connected with the Groups U(n), U(n1,1) and 1U(n1), 12. Representations of the Heisenberg Group and Special Functions, 13. Representations of the Discrete Groups and Special Functions of Discrete Argument.
There are an extensive bibliography (468 titles), list of the most important notations (46 items), and subject index (115 notions).
Representation of Lie Groups and Special Functions is wellwritten book which is a good embellishment of the series "Mathematics and Its Applications" (managing Editor M. Hazewintel). This encyclopedia is the unique, best reference to the foudations, principles, and applications of the special functions from the grouptheoretical viewpoint. I believe, the purchase of this monograph is a good investment for research mathematicians, physicists, engineers, and libraries alike.