Representation of Lie Groups and Special Functions, Softcover reprint of the original 1st ed. 1991
Volume 1: Simplest Lie Groups, Special Functions and Integral Transforms
Mathematics and its Applications Series, Vol. 72

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.

0: Introduction.- 1: Elements of the Theory of Lie Groups and Lie Algebras.- 1.0. Preliminary Information from Algebra, Topology, and Functional Analysis.- 1.1. Lie Groups and Lie Algebras.- 1.2. Homogeneous Spaces with Semisimple Groups of Motions.- 2: Group Representations and Harmonic Analysis on Groups.- 2.1. Representations of Lie Groups and Lie Algebras.- 2.2. Basic Concepts of the Theory of Representations.- 2.3. Harmonic Analysis on Groups and on Homogeneous Spaces.- 3: Commutative Groups and Elementary Functions. The Group of Linear Transformations of the Straight Line and the Gamma-Function. Hypergeometric Functions.- 3.1. Representations of One-Dimensional Commutative Lie Groups and Elementary Functions.- 3.2. The Groups SO(2) and R, Fourier Series and Integrals.- 3.3. Fourier Transform in the Complex Domain. Mellin and Laplace Transforms.- 3.4. Representations of the Group of Linear Transforms of the Straight Line and the Gamma-Function.- 3.5. Hypergeometric Functions and Their Properties.- 4: Representations of the Groups of Motions of Euclidean and Pseudo-Euclidean Planes, and Cylindrical Functions.- 4.1. Representations of the Group ISO(2) and Bessel Functions with Integral Index.- 4.2. Representations of the Group ISO(1,1), Macdonald and Hankel Functions.- 4.3. Functional Relations for Cylindrical Functions.- 4.4. Quasi-Regular Representations of the Groups ISO(2), ISO(1,1) and Integral Transforms.- 5: Representations of Groups of Third Order Triangular Matrices, the Confluent Hypergeometric Function, and Related Polynomials and Functions.- 5.1. Representations of the Group of Third Order Real Triangular Matrices.- 5.2. Functional Relations for Whittaker Functions.- 5.3. Functional Relations for the Confluent Hypergeometric Function and for Parabolic Cylinder Functions.- 5.4. Integrals Involving Whittaker Functions and Parabolic Cylinder Functions.- 5.5. Representations of the Group of Complex Third Order Triangular Matrices, Laguerre and Charlier Polynomials.- 6: Representations of the Groups SU(2), SU(1,1) and Related Special Functions: Legendre, Jacobi, Chebyshev Polynomials and Functions, Gegenbauer, Krawtchouk, Meixner Polynomials.- 6.1. The Groups SU(2) and SU(1,1).- 6.2. Finite Dimensional Irreducible Representations of the Groups GL(2,C) and SU(2).- 6.3. Matrix Elements of the Representations T? of the Group SL(2, C) and Jacobi, Gegenbauer and Legendre Polynomials.- 6.4. Representations of the Group SU(1,1).- 6.5. Matrix Elements of Representations of SU(1, 1), Jacobi and Legendre Functions.- 6.6. Addition Theorems and Multiplication Formulas.- 6.7. Generating Functions and Recurrence Formulas.- 6.8. Matrix Elements of Representations of SU(2) and SU(1,1) as Functions of Column Index. Krawtchouk and Meixner Polynomials.- 6.9. Characters of Representations of SU(2) and Chebyshev Polynomials.- 6.10. Expansion of Functions on the Group SU(2).- 7: Representations of the Groups SU(1,1) and SL(2,?) in Mixed Bases. The Hypergeometric Function.- 7.1. The Realization of Representations T? in the Space of Functions on the Straight Line.- 7.2. Calculation of the Kernels of Representations R?.- 7.3. Functional Relations for the Hypergeometric Function.- 7.4. Special Functions Connected with the Hypergeometric Function.- 7.5. The Mellin Transform and Addition Formulas for the Hypergeometric Function.- 7.6. The Kernels K33(?,?; ?; g) and Hankel Functions.- 7.7. The Kernels Kij(?, ?; ? g), i ? j, and Special Functions.- 7.8. Harmonic Analysis on the Group SL(2, R) and Integral Transforms.- 8: Clebsch-GordanCoefficients, Racah Coefficients, and Special Functions.- 8.1. Clebsch-Gordan Coefficients of the Group SU(2).- 8.2. Properties of CGC’s of the Group SU(2).- 8.3. CGC’s, the Hypergeometric Function 3F2(…; 1) and Jacobi Polynomials.- 8.4. Racah Coefficients of SU(2) and the Hypergeometric Function 4F3(…; 1).- 8.5. Hahn and Racah Polynomials.- 8.6. Clebsch-Gordan and Racah Coefficients of the Group S and Orthogonal Polynomials.- 8.7. Clebsch-Gordan Coefficients of the Group SL(2, R).