Andrews G.E., Askey R., Roy R. / Эндрюс Дж.Э., Аски Р., Рой Р. - Special Functions / Специальные функции [1999, DjVu, ENG]

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Annotation 1
Contents 7
Preface 13
1 The Gamma and Beta Functions 17
1.1 The Gamma and Beta Integrals and Functions 18
1.2 The Euler Reflection Formula 25
1.3 The Hurwitz and Riemann Zeta Functions 31
1.4 Stirling’s Asymptotic Formula 34
1.5 Gauss’s Multiplication Formula for Γ(mx) 38
1.6 Integral Representations for Log Γ(x) and ψ(x) 42
1.7 Kummer’s Fourier Expansion of Log Γ(x) 45
1.8 Integrals of Dirichlet and Volumes of Ellipsoids 48
1.9 The Bohr-Mollerup Theorem 50
1.10 Gauss and Jacobi Sums 52
1.11 A Probabilistic Evaluation of the Beta Function 59
1.12 The p-adic Gamma Function 60
Exercises 62
2 The Hypergeometric Functions 77
2.1 The Hypergeometric Series 77
2.2 Euler’s Integral Representation 81
2.3 The Hypergeometric Equation 89
2.4 The Barnes Integral for the Hypergeometric Function 101
2.5 Contiguous Relations 110
2.6 Dilogarithms 118
2.7 Binomial Sums 123
2.8 Dougall’s Bilateral Sum 125
2.9 Fractional Integration by Parts and Hypergeometric Integrals 127
Exercises 130
3 Hypergeometric Transformations and Identities 140
3.1 Quadratic Transformations 141
3.2 The Arithmetic-Geometric Mean and Elliptic Integrals 148
3.3 Transformations of Balanced Series 156
3.4 Whipple’s Transformation 159
3.5 Dougall’s Formula and Hypergeometric Identities 163
3.6 Integral Analogs of Hypergeometric Sums 166
3.7 Contiguous Relations 170
3.8 The Wilson Polynomials 173
3.9 Quadratic Transformations - Riemann’s View 176
3.10 Indefinite Hypergeometric Summation 179
3.11 The W-Z Method 182
3.12 Contiguous Relations and Summation Methods 190
Exercises 192
4 Bessel Functions and Confluent Hypergeometric Functions 203
4.1 The Confluent Hypergeometric Equation 204
4.2 Barnes’s Integral for 1F1 208
4.3 Whittaker Functions 211
4.4 Examples of 1F1 and Whittaker Functions 212
4.5 Bessel’s Equation and Bessel Functions 215
4.6 Recurrence Relations 218
4.7 Integral Representations of Bessel Functions 219
4.8 Asymptotic Expansions 225
4.9 Fourier Transforms and Bessel Functions 226
4.10 Addition Theorems 229
4.11 Integrals of Bessel Functions 232
4.12 The Modified Bessel Functions 238
4.13 Nicholson’s Integral 239
4.14 Zeros of Bessel Functions 241
4.15 Monotonicity Properties of Bessel Functions 245
4.16 Zero-Free Regions for 1F1 Functions 247
Exercises 250
5 Orthogonal Polynomials 256
5.1 Chebyshev Polynomials 256
5.2 Recurrence 260
5.3 Gauss Quadrature 264
5.4 Zeros of Orthogonal Polynomials 269
5.5 Continued Fractions 272
5.6 Kernel Polynomials 275
5.7 Parseval’s Formula 279
5.8 The Moment-Generating Function 282
Exercises 285
6 Special Orthogonal Polynomials 293
6.1 Hermite Polynomials 294
6.2 Laguerre Polynomials 298
6.3 Jacobi Polynomials and Gram Determinants 309
6.4 Generating Functions for Jacobi Polynomials 313
6.5 Completeness of Orthogonal Polynomials 322
6.6 Asymptotic Behavior of P_n^(α,β)(x) for Large n 326
6.7 Integral Representations of Jacobi Polynomials 329
6.8 Linearization of Products of Orthogonal Polynomials 332
6.9 Matching Polynomials 339
6.10 The Hypergeometric Orthogonal Polynomials 346
6.11 An Extension of the Ultraspherical Polynomials 350
Exercises 355
7 Topics in Orthogonal Polynomials 371
7.1 Connection Coefficients 372
7.2 Rational Functions with Positive Power Series Coefficients 379
7.3 Positive Polynomial Sums from Quadrature and Vietoris’s Inequality 387
7.4 Positive Polynomial Sums and the Bieberback Conjecture 397
7.5 A Theorem of Turin 400
7.6 Positive Summability of Ultraspherical Polynomials 404
7.7 The Irrationality of ζ(3) 407
Exercises 411
8 The Selberg Integral and Its Applications 417
8.1 Selberg’s and Aomoto’s Integrals 418
8.2 Aomoto’s Proof of Selberg’s Formula 418
8.3 Extensions of Aomoto’s Integral Formula 423
8.4 Anderson’s Proof of Selberg’s Formula 427
8.5 A Problem of Stieltjes and the Discriminant of a Jacobi Polynomial 431
8.6 Siegel’s Inequality 435
8.7 The Stieltjes Problem on the Unit Circle 441
8.8 Constant-Term Identities 442
8.9 Nearly Poised 3F2 Identities 444
8.10 The Hasse-Davenport Relation 446
8.11 A Finite-Field Analog of Selberg’s Integral 450
Exercises 455
9 Spherical Harmonics 461
9.1 Harmonic Polynomials 461
9.2 The Laplace Equation in Three Dimensions 463
9.3 Dimension of the Space of Harmonic Polynomials of Degree k 465
9.4 Orthogonality of Harmonic Polynomials 467
9.5 Action of an Orthogonal Matrix 468
9.6 The Addition Theorem 470
9.7 The Funk-Hecke Formula 474
9.8 The Addition Theorem for Ultraspherical Polynomials 475
9.9 The Poisson Kernel and Dirichlet Problem 479
9.10 Fourier Transforms 480
9.11 Finite-Dimensional Representations of Compact Groups 482
9.12 The Group SU(2) 485
9.13 Representations of SU(2) 487
9.14 Jacobi Polynomials as Matrix Entries 489
9.15 An Addition Theorem 490
9.16 Relation of SU(2) to the Rotation Group SO(3) 492
Exercises 494
10 Introduction to q-Series 497
10.1 The q-Integral 501
10.2 The q-Binomial Theorem 503
10.3 The q-Gamma Function 509
10.4 The Triple Product Identity 512
10.5 Ramanujan’s Summation Formula 517
10.6 Representations of Numbers as Sums of Squares 522
10.7 Elliptic and Theta Functions 524
10.8 q-Beta Integrals 529
10.9 Basic Hypergeometric Series 536
10.10 Basic Hypergeometric Identities 539
10.11 q-Ultraspherical Polynomials 543
10.12 Mellin Transforms 548
Exercises 558
11 Partitions 569
11.1 Background on Partitions 569
11.2 Partition Analysis 571
11.3 A Library for the Partition Analysis Algorithm 573
11.4 Generating Functions 575
11.5 Some Results on Partitions 579
11.6 Graphical Methods 581
11.7 Congruence Properties of Partitions 585
Exercises 589
12 Bailey Chains 593
12.1 Rogers’s Second Proof of the Rogers-Ramanujan Identities 593
12.2 Bailey’s Lemma 598
12.3 Watson’s Transformation Formula 602
12.4 Other Applications 605
Exercises 606
A Infinite Products 611
A.l Infinite Products 611
Exercises 613
B Summability and Fractional Integration 615
B.l Abel and Cesaro Means 615
B.2 The Cesaro Means (C,α) 618
B.3 Fractional Integrals 620
B.4 Historical Remarks 621
Exercises 623
C Asymptotic Expansions 627
C.1 Asymptotic Expansion 627
C.2 Properties of Asymptotic Expansions 628
C.3 Watson’s Lemma 630
C.4 The Ratio of Two Gamma Functions 631
Exercises 632
D Euler-Maclaurin Summation Formula 633
D.1 Introduction 633
D.2 The Euler-Maclaurin Formula 635
D.3 Applications 637
D.4 The Poisson Summation Formula 639
Exercises 643
E Lagrange Inversion Formula 645
E.1 Reversion of Series 645
E.2 A Basic Lemma 646
E.3 Lambert’s Identity 647
E.4 Whipple’s Transformation 648
Exercises 650
F Series Solutions of Differential Equations 653
F.l Ordinary Points 653
F.2 Singular Points 654
F.3 Regular Singular Points 655
Bibliography 657
Index 671
Subject Index 675
Symbol Index 677