Representation of Lie groups and special functions: 1 Simplest Lie groups, special functions and integral transforms
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| Sprache: | English |
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Kluwer Academic Publications
1991
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| Schriftenreihe: | Mathematics and its applications. Soviet series
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| Beschreibung: | XXIII, 608 S. |
| ISBN: | 0792314662 0792314948 |
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| 100 | 1 | |a Vilenkin, Naum Ja. |d 1920-1991 |e Verfasser |0 (DE-588)127328122 |4 aut | |
| 245 | 1 | 0 | |a Representation of Lie groups and special functions |n 1 |p Simplest Lie groups, special functions and integral transforms |c by N. Ja. Vilenkin and A. U. Klimyk |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer Academic Publications |c 1991 | |
| 300 | |a XXIII, 608 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its applications. Soviet series |v 72 | |
| 490 | 0 | |a Mathematics and its Applications / Soviet Series |v ... | |
| 700 | 1 | |a Klimyk, Anatolij U. |d 1939-2008 |e Verfasser |0 (DE-588)115774580 |4 aut | |
| 773 | 0 | 8 | |w (DE-604)BV005437238 |g 1 |
| 830 | 0 | |a Mathematics and its applications. Soviet series |v 72 |w (DE-604)BV004708148 |9 72 | |
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Datensatz im Suchindex
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| adam_text | Table of Contents
Series Editor s Preface v
Preface xix
List of Special Symbols Xxi
Chapter 0: Introduction 1
Chapter 1:
Elements of the Theory of Lie Groups and Lie
Algebras 6
1.0. Preliminary Information from Algebra, Topology, and
Functional Analysis 6
1.0.1. Groups, rings, linear spaces, Lie algebras 6
1.0.2. Subgroups and subalgebras 9
1.0.3. Homomorphisms and automorphisms 11
1.0.4. Direct products, direct sums, multilinear functionals .... 15
1.0.5. The matrix form of linear operators, bilinear
and sesquilinear functionals 18
1.0.6. Topological spaces and manifolds 19
1.0.7. Generalized functions 22
1.0.8. Banach and Hilbert spaces 25
1.0.9. Countably Hilbert spaces and nuclear spaces 28
1.0.10. Eigenvalues and eigenvectors 30
1.0.11. Tensor product of spaces and operators 32
1.0.12. Banach algebras 34
1.1. Lie Groups and Lie Algebras 35
1.1.1. Linear groups 35
1.1.2. Lie groups 38
1.1.3. Correspondence between Lie groups and Lie algebras ... 40
1.1.4. The exponential mapping 42
1.1.5. The adjoint Lie group. The center of the universal
enveloping algebra 45
1.1.6. The Killing form 46
1.1.7. Semisimple Lie algebras and groups 48
1.1.8. Nilpotent and solvable Lie algebras and groups 49
1.2. Homogeneous Spaces with Semisimple Groups of Motions . 51
1.2.1. Homogeneous spaces 51
1.2.2. Symmetric homogeneous spaces 55
1.2.3. Cartan decomposition of semisimple Lie algebras
and homogeneous spaces with semisimple
motion groups 56
viii Table of Contents
1.2.4. Pseudo Riemannian symmetric homogeneous spaces .... 59
1.2.5. Complex homogeneous domains 61
1.2.6. Invariant measures 62
1.2.7. The group ring of a Lie group 66
Chapter 2:
Group Representations and Harmonic Analysis
on Groups 68
2.1. Representations of Lie Groups and Lie Algebras 68
2.1.1. Group representations 68
2.1.2. Expressions for representations in matrix form 71
2.1.3. Representations of group rings 73
2.1.4. Infinite differentiability of matrix elements of
representations of Lie groups 74
2.1.5. Representations of Lie algebras and of
universal enveloping algebras 75
2.1.6. Group representations by shift operators in
function spaces 77
2.2. Basic Concepts of the Theory of Representations 80
2.2.1. Invariant subspaces. Cyclic, irreducible and
completely reducible representations 80
2.2.2. Contragradient representations 82
2.2.3. Unitary representations 83
2.2.4. Intertwining and invariant operators 85
2.2.5. Equivalent representations 87
2.2.6. Representations of complex Lie groups and algebras and of
their real forms 88
2.2.7. Characters of finite dimensional representations 89
2.2.8. Schur s lemma 90
2.2.9. Invariant bilinear and sesquilinear functionals 91
2.2.10. Intertwining operators for completely reducible
representations 93
2.3. Harmonic Analysis on Groups and on Homogeneous Spaces 94
2.3.1. Introduction 94
2.3.2. The realization of irreducible representations by
shift operators 95
2.3.3. One sided invariant subspaces of VJIt 96
2.3.4. The Peter Weyl theorem 97
2.3.5. The complete orthonormal system of functions on a
compact group 99
2.3.6. Decomposition of the regular representation of a
compact group 101
I
Table of Contents jx
2.3.7. The approximation theorem 102
2.3.8. Spherical functions 103
2.3.9. Expansion of functions on homogeneous
compact spaces 104
2.3.10. Expansion of functions on a compact group and
the group ring 104
2.3.11. Expansion of central functions 107
Chapter 3:
Commutative Groups and Elementary Functions. The
Group of Linear Transformations of the Straight Line
and the Gamma Function. Hypergeometric Functions 109
3.1. Representations of One Dimensional Commutative Lie
Groups and Elementary Functions 109
3.1.1. Irreducible representations of the additive group of
real numbers and the exponential function 109
3.1.2. Representations of the multiplicative group of
positive numbers and the power function 109
3.1.3. Trigonometric functions and representations of
the group T 110
3.1.4. The group of hyperbolic rotations of the plane and
hyperbolic functions Ill
3.1.5. The connection between the groups 50(2), 50(1,1)
and 50(2, C) 112
3.1.6. The generalized functions x+, xt, x x, x x
sign x, (x ± i0)x and their properties 112
3.2. The Groups 50(2) and R, Fourier Series and Integrals . . 114
3.2.1. Fourier Series as expansion of functions, defined on 50(2) 114
3.2.2. Expansion of infinitely differentiable functions 115
3.2.3. The Fourier transform 116
3.2.4. Decomposition of the regular representation of R .... 119
3.3. Fourier Transform in the Complex Domain.
Mellin and Laplace Transforms 120
3.3.1. Fourier transform in the complex domain 120
3.3.2. The Laplace transform 122
3.3.3. Transformation of square integrable functions 122
3.3.4. The Mellin transform 124
3.3.5. The Gauss Weierstrass transform 126
3.4. Representations of the Group of Linear Transforms of the
Straight Line and the Gamma Function 127
3.4.1. The group of linear transformations of the straight line
and its representations 127
x Table of Contents
3.4.2. Diagonalization of the operators R (g(a,0)) 129
3.4.3. Expression for the kernel K(w,z;g) in terms of
the F function 131
3.4.4. Properties of the F function 132
3.4.5. The addition formula for the F function and
its corollaries 135
3.4.6. The beta function and the multiplication formula
forlXz) 137
3.4.7. The Fourier transform of the functions x and x _ ... 139
3.5. Hypergeometric Functions and Their Properties 141
3.5.1. Hypergeometric functions 141
3.5.2. Some properties of hypergeometric functions 142
3.5.3. Elementary properties of the hypergeometric
function 2 F1 145
3.5.4. Some integrals involving the hypergeometric function . . 147
3.5.5. Dougall s formula 148
3.5.6. Cylindrical functions 148
3.5.7. Whittaker functions, parabolic cylinder function, Laguerre
and Hermite polynomials 155
3.5.8. Jacobi and Legendre polynomials and functions 156
3.5.9. Functions of discrete variable 157
3.5.10. Fractional integration 160
3.5.11. Fractional integration and special functions 161
Chapter 4:
Representations of the Groups of Motions of Euclidean
and Pseudo Euclidean Planes, and
Cylindrical Functions 165
4.1. Representations of the Group 750(2) and Bessel Functions
with Integral Index 165
4.1.1. The Group 750(2) 165
4.1.2. Irreducible representations of 750(2) 166
4.1.3. Calculation of matrix elements of the representations T/j . 167
4.1.4. The addition theorem and the multiplication formula . . 168
4.1.5. The generating function 170
4.2. Representations of the Group 750(1,1), Macdonald
and Hankel Functions 171
4.2.1. The pseudo Euclidean plane 171
4.2.2. The group 750(1,1) 171
4.2.3. Representations of 750(1,1) 173
4.2.4. The representations Tr, Macdonald and
Hankel functions 175
Table of Contents xi
4.3. Functional Relations for Cylindrical Functions 179
4.3.1. Introductory remarks 179
4.3.2. Recurrence relations and differential equations 180
4.3.3. Cylindrical functions with half integral index 181
4.3.4. Integral representations of the Barnes type 182
4.3.5. Mellin transforms 184
4.3.6. Addition theorems 188
4.3.7. Multiplication theorems 191
4.3.8. The Ramanujan formula 192
4.4. Quasi Regular Representations of the Groups JSO(2),
ISO(1,1) and Integral Transforms 193
4.4.1. The quasi regular representation of the group ISO(2) . . 193
4.4.2. The Fourier Bessel transform 196
4.4.3. The quasi regular representation of/5O(l, 1) 198
4.4.4. Integral transforms 200
4.4.5. Mutually reciprocal integral transform 204
Chapter 5:
Representations of Groups of Third Order Triangular
Matrices, the Confluent Hypergeometric Function,
and Related Polynomials and Functions 207
5.1. Representations of the Group of Third Order Real
Triangular Matrices 207
5.1.1. The group of third order triangular matrices 207
5.1.2. Irreducible representations of the group G 208
5.1.3. Another realization of the representations Tx 209
5.1.4. Calculation of the kernels of the representations Rx . . . 211
5.2. Functional Relations for Whittaker Functions 213
5.2.1. Relations between infinitesimal operators and operators
of the representation Rx 213
5.2.2. Recurrence relations 214
5.2.3. The Whittaker differential equation 217
5.2.4. The Mellin Barnes integral representation 219
5.2.5. Symmetry relations for Whittaker functions 221
5.3. Functional Relations for the Confluent Hypergeometric
Function and for Parabolic Cylinder Functions 221
5.3.1. The functions $(a; 7; z) and $(a; 7; z) 221
5.3.2. Recurrence relations for $(a; 7; z) and ^(a; 7; 2) .... 222
5.3.3. The differential equation 225
5.3.4. The connection of the functions $(a; 7; z), *( *; 7; z),
M n(z), V piz) with cylindrical functions 225
5.3.5. Parabolic cylinder functions 227
xii Table of Contents
5.3.6. Hermite polynomials 232
5.4. Integrals Involving Whittaker Functions and Parabolic
Cylinder Functions 236
5.4.1. The Mellin transform in parameters 236
5.4.2. Continuous addition theorems 239
5.4.3. Multiplication theorems 242
5.4.4. Degenerate cases of addition theorems 243
5.4.5. Integral transforms connected with parabolic cylinder
functions and Whittaker functions 244
5.5. Representations of the Group of Complex Third Order
Triangular Matrices, Laguerre and Charlier Polynomials . 247
5.5.1. Representations of the group of complex third order
triangular matrices 247
5.5.2. Recurrence relations and the differential equation for
Laguerre polynomials 252
5.5.3. The Rodrigues formula and generating functions
for Laguerre polynomials 254
5.5.4. The orthogonality relation for Laguerre polynomials . . . 256
5.5.5. Summation formulas for Laguerre polynomials 257
5.5.6. Addition theorems and multiplication formulas for
Laguerre polynomials 261
5.5.7. The connection btween Laguerre polynomials and
cylindrical functions 264
5.5.8. Charlier polynomials 264
Chapter 6:
Representations of the Groups SU (2), 5(7(1,1) and
Related Special Functions: Legendre, Jacobi,
Chebyshev Polynomials and Functions, Gegenbauer,
Krawtchouk, Meixner Polynomials 269
6.1. The Groups 5*7(2) and 51/(1,1) 269
6.1.1. Parametrization 269
6.1.2. Lie algebras of the groups 5*7(2) and SU( 1,1) 272
6.1.3. Connection with other groups 273
6.1.4. Invariant measures 274
6.1.5. The universal covering group for SU(1,1) 275
6.2. Finite Dimensional Irreducible Representations of the
Groups GL(2, C) and SU{2) 276
6.2.1. Representations in spaces of homogeneous polynomials . . 276
6.2.2. Infinitesimal operators of the representation Ti 277
6.2.3. The invariant scalar product 278
Table of Contents xijj
6.3. Matrix Elements of the Representations Tt of the Group
SL(2, C) and Jacobi, Gegenbauer and Legendre Polynomials 280
6.3.1. Calculation of matrix elements 280
6.3.2. Other expressions for the matrix elements 283
6.3.3. Expressions in terms of the Euler angles 284
6.3.4. Expression for the function P^n{z) in terms of
the hypergeometric series 286
6.3.5. Integral representations, the Rodrigues formula
for Pin(z) 287
6.3.6. Symmetry relations and special values of P^n(z) .... 287
6.3.7. Connection of P^n(z) with the classical orthogonal
polynomials 288
6.3.8. Some properties of Jacobi polynomials 289
6.3.9. Gegenbauer polynomials 291
6.3.10. Legendre polynomials 295
6.3.11. Legendre polynomials as zonal spherical functions .... 297
6.3.12. Bessel functions and Jacobi polynomials 297
6.4. Representations of the Group 517(1,1) 298
6.4.1. The representations Tx 298
6.4.2. Irreducibility at non integral points 300
6.4.3. Representations with integral x 301
6.4.4. Equivalent representations 303
6.4.5. Hermitian adjoint representations 306
6.4.6. Unitary representations 307
6.4.7. Irreduciblejmitary representations of the
group 517(1,1) 310
6.5. Matrix Elements of Representations of 5f7(l,l), Jacobi and
Legendre Functions 311
6.5.1. Calculation of the matrix elements of the
representations Tx 311
6.5.2. Expressions for matrix elements in terms of Euler angles . 312
6.5.3. Expression for ?P^n(cosh t) in terms of the
hypergeometric function 313
6.5.4. Integral representations of ^3jnn(cosh t) 314
6.5.5. Symmetry relations for ^}^n(cosh t) 318
6.5.6. The functions $P^n(cosh t) in the integral case 320
6.5.7. Connection of ^J^n(cosh t) with special functions .... 322
6.5.8. The Rodrigues formula for the functions ^J^n(cosh t) . . 323
6.5.9. Legendre functions and associated Legendre functions . . 324
6.5.10. Zonal and associated spherical functions of the
representations Tx 325
xiv Table of Contents
6.6. Addition Theorems and Multiplication Formulas 326
6.6.1. Addition theorems for the functions ?Pjnn(cosh t) . . . . 326
6.6.2. Addition theorems for Legendre functions 327
6.6.3. Multiplication formulas 327
6.6.4. Analog of the Ramanujan Formula 330
6.7. Generating Functions and Recurrence Formulas 331
6.7.1. Generating functions for fixed £, r and n 331
6.7.2. Recurrence formulas for Pmn(z) and ^Pmn(cos^ 0
for fixed I and r 334
6.7.3. Recurrence relations for P^n(z) and ?P£,n(cosh t)
with various I and r 336
6.7.4. Recurrence relations for Jacobi polynomials 338
6.7.5. The differential equations for P£n(z), PiaJ)(z)
and^O*) 341
6.7.6. The differential equation and recurrence relations for
Gegenbauer polynomials 342
6.7.7. Recurrence relations for Legendre polynomials 344
6.7.8. The generating function for P^ni2) f°r fixed m an(l n ¦ • 344
6.7.9. The continuous generating function for $Pjj,n(cosh t) . . . 346
6.8. Matrix Elements of Representations of SU(2) and SU(1,1) as
Functions of Column Index. Krawtchouk and Meixner
Polynomials 346
6.8.1. The representations Tt of SU(2) and Krawtchouk
polynomials 346
6.8.2. The discrete series representations of the group
SU(1,1) and Meixner polynomials 350
6.8.3. Connection of Krawtchouk and Meixner polynomials with
other polynomials 353
6.8.4. Krawtchouk Meixner functions 354
6.9. Characters of Representations of SU(2) and Chebyshev
Polynomials 357
6.9.1. Chebyshev polynomials 357
6.9.2. Calculation of characters of the representations Tf
of the group SU(2) 358
6.9.3. Characters and Chebyshev polynomials 360
6.9.4. Expansion of functions in Chebyshev polynomials and the
completeness of the system of the representations Te . . 361
6.9.5. The tensor product of the representations Tt of SU(2) . . 363
6.10. Expansion of Functions on the Group 517(2) 364
6.10.1. Orthogonality relations for the functions P^n(z) and
related polynomials 364
Table of Contents xv
6.10.2. Expansions into series in P^n^2) an ^ m
related polynomials 366
6.10.3. The Laplace operator 370
6.10.4. Expansion of infinitely differentiable functions 372
Chapter 7:
Representations of the Groups SU (1,1) and SX(2,R) in
Mixed Bases. The Hypergeometric Function 374
7.1. The Realization of Representations Tx in the Space of
Functions on the Straight Line 374
7.1.1. Parametrization of 5L(2, R) 374
7.1.2. A new realization of representations Tx 375
7.1.3. Bases of the space Tx 376
7.1.4. The representation Rx 378
7.1.5. Infinitesimal operators of representations Rx 380
7.2. Calculation of the Kernels of Representations Rx 381
7.2.1. Calculation of K22(X,fj.;x;h) and K22{ ,n; ;u) .... 381
7.2.2. The kernels K22(A,/i;x;?) for triangular matrices . . . 384
7.3. Functional Relations for the Hypergeometric Function . . 386
7.3.1. Relations between infinitesimal operators and
representation operators 386
7.3.2. Recurrence formulas 387
7.3.3. The hypergeometric equation and its solutions 390
7.3.4. Integral representations for the hypergeometric function . 391
7.3.5. Formulas of linear transformations 396
7.3.6. Formulas of quadratic transformations 397
7.4. Special Functions Connected with the Hypergeometric
Function 398
7.4.1. The functions $TXll(z) and Q^(z) 398
7.4.2. Expressions for the kernels A 22(A, fi; x; h) in terms of the
functions ^(z) and Q^(z) 401
7.4.3. Jacobi functions 401
7.4.4. Associated Legendre functions of the first and of the
second kinds 404
7.4.5. Legendre functions of the second kind 407
7.4.6. Gegenbauer functions 408
7.5. The Mellin Transform and Addition Formulas for the
Hypergeometric Function 409
7.5.1. The Mellin transform 409
7.5.2. Mellin transform (degenerate cases) 413
7.5.3. Addition theorems 416
7.5.4. Degenerate cases of addition theorems 419
xvi Table of Contents
7.5.5. Integral transforms, connected with the
hypergeometric function 422
7.6. The Kernels K33(X,/i;x;g) and Hankel Functions 423
7.6.1. The realization of representations Tx, connected with
the subgroup of triangular matrices 423
7.6.2. Calculation of the kernel of Qx(s) 424
7.6.3. The kernels of Qx(g) 426
7.6.4. Mutually reciprocal integral transforms 427
7.6.5. Functional relations for cylindrical functions 428
7.6.6. Addition theorems for H^)+1(x) and K2t {x) 430
7.7. The Kernels Kij( ,fi;x;g), *V.? and Special Functions . . 434
7.7.1. Integral representations of the kernels K i( ,ii;x;g) . . . 434
7.7.2. Some properties of the kernels 437
7.7.3. Calculation of the kernels 439
7.7A. Mutually reciprocal integral transforms 446
7.7.5. The Mellin transform 447
7.7.6. Addition theorems 450
7.7.7. Discrete series representations of the group SL(2, R) . . . 458
7.7.8. Discrete series representations in the oscillator form . . . 465
7.7.9. Discrete series representations and special functions . . . 468
7.7.10. Integral transforms 471
7.7.11. Pollaczek Meixner polynomials 473
7.8. Harmonic Analysis on the Group SL(2,R) and Integral
Transforms 475
7.8.1. Fourier transform of functions on the group SX(2, R) . . . 475
7.8.2. Characters of irreducible representations of SL(2, R) . . . 476
7.8.3. Derivation of the inversion formula 478
7.8.4. Integral transforms, connected with the Fourier transform
on 51(2, R) 483
7.8.5. Decomposition of representations 488
7.8.6. The Laplace operator on the group SL(2, R) 489
7.8.7. Eigenfunctions of the Laplace operator on SL(2, R) ... 492
7.8.8. The inversion formula for the Jacobi transform 493
Chapter 8:
Clebsch Gordan Coefficients, Racah Coefficients,
and Special Functions 498
8.1. Clebsch Gordan Coefficients of the Group SU(2) 498
8.1.1. Realization of the tensor product Ttx g) T 2 in the
space of homogeneous polynomials 498
8.1.2. Clebsch Gordan coefficients (CGC s) of the group SU(2) . 500
8.1.3. Calculation of CGC s 501
Table of Contents xvii
8.1.4. Expression for matrix elements of the representation
T( in terms of CGC s 503
8.2. Properties of CGC s of the Group SU(2) 504
8.2.1. Generating functions 504
8.2.2. Symmetry relations 505
8.2.3. Relations between CGC s and matrix elements of irreducible
representations 506
8.2.4. Integral representations 507
8.2.5. Expressions for CGC s in the form of finite sums .... 509
8.2.6. Special values of CGC s 511
8.2.7. Recurrence relations and difference equations for CGC s . 513
8.2.8. Addition theorems 516
8.3. CGC s, the Hypergeometric Function 3 F2(...; 1) and
Jacobi Polynomials 517
8.3.1. CGC s and the hypergeometric function 3i 2(...; 1) . . . 517
8.3.2. Symmetry relations for the hypergeometric
, function 3F2(...;1) 520
8.3.3. Summation formulas for hypergeometric series 521
8.3.4. Asymptoptic formulas 523
8.3.5. Recurrence relations and difference equations for the function
3*2(...;1) 524
8.3.6. CGC s and products of Jacobi polynomials 526
8.3.7. CGC s and new recurrence formulas for the
functions P^n{z) 527
8.3.8. The Burchnall Chaundy formula 527
8.4. Racah Coefficients of SU(2) and the Hypergeometric
Function 4F3(...;1) 529
8.4.1. Definition of Racah coefficients (RC s) 529
8.4.2. Connection between RC s and CGC s 532
8.4.3. Symmetry relations for RC s 533
8.4.4. RC s and the hypergeometric function iF$(...; 1) . . . . 534
8.4.5. Special cases of RC s 536
8.4.6. Expressions for RC s in terms of characters
of representations 537
8.4.7. The addition theorem for RC s 538
8.4.8. The Biedenharn Elliott identity 539
8.4.9. Recurrence relations for RC s 540
8.4.10. CGC as a limit of RC 541
8.4.11. New addition theorems for CGC s 542
8.4.12. Symmetry relations for 4F3(...;1) 543
8.4.13. The Whipple formula and its consequences 543
xvjjj Table of Contents
8.5. Hahn and Racah Polynomials 546
8.5.1. CGC s and Hahn polynomials 546
8.5.2. Dual Hahn polynomials (Eberlane polynomials) 549
8.5.3. The multiplication formula and the addition theorem for
Krawtchouk polynomials 550
8.5.4. Racah polynomials 552
8.5.5. Wilson polynomials 556
8.5.6. Addition Theorems for Hahn and Racah polynomials . . 558
8.5.7. Hahn polynomials as a limit of Racah and
Wilson polynomials 559
8.6. Clebsch Gordan and Racah Coefficients of the Group S
and Orthogonal Polynomials 561
8.6.1. The tensor product of representations of the group S . . 561
8.6.2. Clebsch Gordan coefficients 563
8.6.3. The generating function and symmetry relations .... 566
8.6.4. Clebsch Gordan coefficients and Laguerre polynomials . . 567
8.6.5. The multiplication formula and the addition theorem
for Charlier polynomials 568
8.6.6. Racah coefficients (RC s) 569
8.7. Clebsch Gordan Coefficients of the Group SL(2, R) 572
8.7.1. CGC s of the tensor product of infinite and
finite dimensional representations 572
8.7.2. Expansion of products of functions ^Pmn(z) and Pmn(z) • ^76
8.7.3. Recurrence relations for ?P^n(cosh t) 577
8.7.4. The tensor product of discrete series representations . . . 578
8.7.5. CGC s for continuous bases 580
8.7.6. CGC s for discrete series representations and
special functions 583
8.7.7. Other tensor products 587
8.7.8. CGC s for the tensor product fXl ® fX2 588
8.7.9. CGC s for the principal unitary series representations . . 590
Bibliography 595
Subject Index 599
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| any_adam_object | 1 |
| author | Vilenkin, Naum Ja. 1920-1991 Klimyk, Anatolij U. 1939-2008 |
| author_GND | (DE-588)127328122 (DE-588)115774580 |
| author_facet | Vilenkin, Naum Ja. 1920-1991 Klimyk, Anatolij U. 1939-2008 |
| author_role | aut aut |
| author_sort | Vilenkin, Naum Ja. 1920-1991 |
| author_variant | n j v nj njv a u k au auk |
| building | Verbundindex |
| bvnumber | BV005437240 |
| classification_rvk | SK 340 SK 680 |
| ctrlnum | (OCoLC)311619658 (DE-599)BVBBV005437240 |
| dewey-full | 512/.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.2 |
| dewey-search | 512/.2 |
| dewey-sort | 3512 12 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV005437240 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T16:29:30Z |
| institution | BVB |
| isbn | 0792314662 0792314948 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-003401095 |
| oclc_num | 311619658 |
| open_access_boolean | |
| owner | DE-384 DE-91G DE-BY-TUM DE-703 DE-824 DE-29T DE-188 |
| owner_facet | DE-384 DE-91G DE-BY-TUM DE-703 DE-824 DE-29T DE-188 |
| physical | XXIII, 608 S. |
| publishDate | 1991 |
| publishDateSearch | 1991 |
| publishDateSort | 1991 |
| publisher | Kluwer Academic Publications |
| record_format | marc |
| series | Mathematics and its applications. Soviet series |
| series2 | Mathematics and its applications. Soviet series Mathematics and its Applications / Soviet Series |
| spelling | Vilenkin, Naum Ja. 1920-1991 Verfasser (DE-588)127328122 aut Representation of Lie groups and special functions 1 Simplest Lie groups, special functions and integral transforms by N. Ja. Vilenkin and A. U. Klimyk Dordrecht [u.a.] Kluwer Academic Publications 1991 XXIII, 608 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications. Soviet series 72 Mathematics and its Applications / Soviet Series ... Klimyk, Anatolij U. 1939-2008 Verfasser (DE-588)115774580 aut (DE-604)BV005437238 1 Mathematics and its applications. Soviet series 72 (DE-604)BV004708148 72 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003401095&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Vilenkin, Naum Ja. 1920-1991 Klimyk, Anatolij U. 1939-2008 Representation of Lie groups and special functions Mathematics and its applications. Soviet series |
| title | Representation of Lie groups and special functions |
| title_auth | Representation of Lie groups and special functions |
| title_exact_search | Representation of Lie groups and special functions |
| title_full | Representation of Lie groups and special functions 1 Simplest Lie groups, special functions and integral transforms by N. Ja. Vilenkin and A. U. Klimyk |
| title_fullStr | Representation of Lie groups and special functions 1 Simplest Lie groups, special functions and integral transforms by N. Ja. Vilenkin and A. U. Klimyk |
| title_full_unstemmed | Representation of Lie groups and special functions 1 Simplest Lie groups, special functions and integral transforms by N. Ja. Vilenkin and A. U. Klimyk |
| title_short | Representation of Lie groups and special functions |
| title_sort | representation of lie groups and special functions simplest lie groups special functions and integral transforms |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003401095&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV005437238 (DE-604)BV004708148 |
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