One service mathematici has rendered the 'Et moi, ... si j'avait IU comment en revenir. je n'y serais point alle.' human race. It has put common sense back Jules Verne where it belong., on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense', Eric T. Bell able to do something with it. O. H eaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other pans and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'el;re of this series."

This is the first of three major volumes which present a comprehensive treatment of the theory of the main classes of special functions from the point of view of the theory of group representations. This volume deals with the properties of classical orthogonal polynomials and special functions which are related to representations of groups of matrices of second order and of groups of triangular matrices of third order. This material forms the basis of many results concerning classical special functions such as Bessel, MacDonald, Hankel, Whittaker, hypergeometric, and confluent hypergeometric functions, and different classes of orthogonal polynomials, including those having a discrete variable. Many new results are given. The volume is self-contained, since an introductory section presents basic required material from algebra, topology, functional analysis and group theory. For research mathematicians, physicists and engineers.

The author describes the current state of the art in the theory of invariant random fields. This theory is based on several different areas of mathematics, including probability theory, differential geometry, harmonic analysis, and special functions. The present volume unifies many results scattered throughout the mathematical, physical, and engineering literature, as well as it introduces new results from this area first proved by the author. The book also presents many practical applications, in particular in such highly interesting areas as approximation theory, cosmology and earthquake engineering. It is intended for researchers and specialists working in the fields of stochastic processes, statistics, functional analysis, astronomy, and engineering.

This is the last of three major volumes which present a comprehensive treatment of the theory of the main classes of special functions from the point of view of the theory of group representations. This volume deals with q-analogs of special functions, quantum groups and algebras (including Hopf algebras), and (representations of) semi-simple Lie groups. Also treated are special functions of a matrix argument, representations in the Gel'fand-Tsetlin basis, and, finally, modular forms, theta-functions and affine Lie algebras. The volume builds upon results of the previous two volumes, and presents many new results. Subscribers to the complete set of three volumes will be entitled to a discount of 15%.

This work aims to give a systematic presentation of methods used in the spectral theory of nonself-adjoint, generally unbounded, operators. Subjects treated include the wide class of both selfadjoint and nonself-adjoint extensions of Hermitian operators; characteristic functions of a regular extension; the construction of some operator models for different classes of nonself-adjoint operators; the construction of the selfadjoint dilation of an arbitrary dissipative operator and J-unitary and J-selfadjoint dilations of linear operators; the abstract Lax Phillips scheme in scattering theory for spaces with indefinite metric (Pontryagin spaces); the relation between scattering matrix and the characteristic function of J-nonexpansive operators; a structure of J-nonexpansive operators; and contractions and their triangulation.

In theoretical physics, routine use is made of many properties, such as recurrence relations and addition theorems, of the special functions of mathematical physics. These properties are for the most part classical, and their derivations are usually based on the methods of classical analysis. The purpose of this book is to show how these functions are also related to the theory of group representations and to derive their important properties from this theory. This approach elucidates the geometric background for the existence of the relations among the special functions. Moreover, the derivations may be more rationally motivated than are the usual complicated manipulations of power series, integral representations, and so on. I hope that the reader may find in this book reasonably simple derivations of many of the relations commonly used in theoretical physics for which the proofs may otherwise be somewhat unfamiliar. In order that the book be fairly self-contained, approximately the first third delves into a preliminary discussion of such topics as Lie groups, group representations, and so on. The remaining chapters are devoted to various groups, and the special functions are discussed in conjunction with the group with which it is associated. Because of the inclusion of the introductory material, the only prerequisite is a reasonable knowledge of linear algebra. The original impetus for the writing of this book was provided by a lecture course given by Professor Eugene P. Wigner a number of years ago. I am greatly indebted to Professor Wigner for his suggestion that I pursue the subject of the lectures further and for his continued friendly interest and advice in the work.-I wish to thank Dr. Trevor Luke for carefully checking the manuscript. I also wish to thank my wife, whose encouragement contributed greatly to the writing of the book.