Special Functions

Preface
page xiii
1
The Gamma and Beta Functions
1
1.1
The Gamma and Beta Integrals and Functions
2
1.2
The Euler Reflection Formula
9
1.3
The Hurwitz and Riemann Zeta Functions
15
1.4
Stirling\'s Asymptotic Formula
18
1.5
Gauss\'s Multiplication Formula for Y(mx)
22
1.6
Integral Representations for Log F(x) and x//(x)
26
1.7
Kummer\'s Fourier Expansion of Log V (x)
29
1.8
Integrals of Dirichlet and Volumes of Ellipsoids
32
1.9
The Bohr-Mollerup Theorem
34
1.10
Gauss and Jacobi Sums
36
1.11
A Probabilistic Evaluation of the Beta Function
43
1.12
The /7-adic Gamma Function
44
Exercises
46
2
The Hypergeometric Functions
61
2.1
The Hypergeometric Series
61
2.2
Euler\'s Integral Representation
65
2.3
The Hypergeometric Equation
73
2.4
The Barnes Integral for the Hypergeometric Function
85
2.5
Contiguous Relations
94
2.6
Dilogarithms
102
2.7
Binomial Sums
107
2.8
Dougall\'s Bilateral Sum
109
2.9
Fractional Integration by Parts and Hypergeometric Integrals
111
Exercises
114
3
Hypergeometric Transformations and Identities
124
3.1
Quadratic Transformations
125
3.2
The Arithmetic-Geometric Mean and Elliptic Integrals
132
3.3
Transformations of Balanced Series
140
3.4
Whipple\'s Transformation
143
3.5
Dougall\'s Formula and Hypergeometric Identities
147
3.6
Integral Analogs of Hypergeometric Sums
150
3.7
Contiguous Relations
154
3.8
The Wilson Polynomials
157
3.9
Quadratic Transformations - Riemann\'s View
160
3.10 Indefinite Hypergeometric Summation
163
3.11 The W-Z Method
166
3.12 Contiguous Relations and Summation Methods
174
Exercises
176
4
Bessel Functions and Confluent Hypergeometric Functions
187
4.1
The Confluent Hypergeometric Equation
188
4.2
Barnes\'s Integral for XFX
192
4.3
Whittaker Functions
195
4.4
Examples of i F\\ and Whittaker Functions
196
4.5
Bessel\'s Equation and Bessel Functions
199
4.6
Recurrence Relations
202
4.7
Integral Representations of Bessel Functions
203
4.8
Asymptotic Expansions
209
4.9
Fourier Transforms and Bessel Functions
210
4.10 Addition Theorems
213
4.11 Integrals of Bessel Functions
216
4.12 The Modified Bessel Functions
222
4.13 Nicholson\'s Integral
223
4.14 Zeros of Bessel Functions
225
4.15 Monotonicity Properties of Bessel Functions
229
4.16 Zero-Free Regions for i F\\ Functions
231
Exercises
234
5
Orthogonal Polynomials
240
5.1
Chebyshev Polynomials
240
5.2
Recurrence
244
5.3
Gauss Quadrature
248
5.4
Zeros of Orthogonal Polynomials
253
5.5
Continued Fractions
256
5.6
Kernel Polynomials
259
5.7
Parseval\'s Formula
263
5.8
The Moment-Generating Function
266
Exercises
269
6
Special Orthogonal Polynomials
277
6.1
Hermite Polynomials
278
6.2
Laguerre Polynomials
282
6.3
Jacobi Polynomials and Gram Determinants
293
6.4
Generating Functions for Jacobi Polynomials
297
6.5
Completeness of Orthogonal Polynomials
306
6.6
Asymptotic Behavior of P^(x)
for Large n
310
6.7
Integral Representations of Jacobi Polynomials
313
6.8
Linearization of Products of Orthogonal Polynomials
316
6.9
Matching Polynomials
323
6.10
The Hypergeometric Orthogonal Polynomials
3 30
6.11
An Extension of the Ultraspherical Polynomials
3 34
Exercises
339
7
Topics in Orthogonal Polynomials
355
7.1
Connection Coefficients
356
7.2
Rational Functions with Positive Power Series Coefficients
363
7.3
Positive Polynomial Sums from Quadrature
and Vietoris \'s Inequality
371
7.4
Positive Polynomial Sums and the Bieberback Conjecture
381
7.5
A Theorem of Turan
384
7.6
Positive Summability of Ultraspherical Polynomials
388
7.7
The Irrationality of ? (3)
391
Exercises
395
8
The Selberg Integral and Its Applications
401
8.1
Selberg\'s and Aomoto\'s Integrals
402
8.2
Aomoto\'s Proof of Selberg\'s Formula
402
8.3
Extensions of Aomoto\'s Integral Formula
407
8.4
Anderson\'s Proof of Selberg\'s Formula
411
8.5
A Problem of Stieltjes and the Discriminant of
a Jacobi Polynomial
415
8.6
Siegel\'s Inequality
419
8.7
The Stieltjes Problem on the Unit Circle
425
8.8
Constant-Term Identities
426
8.9
Nearly Poised 3 F2 Identities
428
8.10
The Hasse-Davenport Relation
430
8.11
A Finite-Field Analog of Selberg s Integral
Exercises
Spherical Harmonics
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
9.10
9.11
9.12
9.13
9.14
9.15
9.16
Harmonic Polynomials
The Laplace Equation in Three Dimensions
Dimension of the Space of Harmonic Polynomials
of Degree k
Orthogonality of Harmonic Polynomials
Action of an Orthogonal Matrix
The Addition Theorem
The Funk-Hecke Formula
The Addition Theorem for Ultraspherical Polynomials
The Poisson Kernel and Dirichlet Problem
Fourier Transforms
Finite-Dimensional Representations of Compact Groups
The Group SU(2)
Representations of SU(2)
Jacobi Polynomials as Matrix Entries
An Addition Theorem
Relation of SU(2) to the Rotation Group SO(3)
Exercises
Introduction to ^r-Series
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
10.9
10.10
10.11
10.12
The q -Integral
The g-Binomial Theorem
The q -Gamma Function
The Triple Product Identity
Ramanujan\'s Summation Formula
Representations of Numbers as Sums of Squares
Elliptic and Theta Functions
g-Beta Integrals
Basic Hypergeometric Series
Basic Hypergeometric Identities
g-Ultraspherical Polynomials
Mellin Transforms
Exercises
Partitions
11.1
11.2
11.3
Background on Partitions
Partition Analysis
A Library for the Partition Analysis Algorithm
434
439
445
445
447
449
451
452
454
458
459
463
464
466
469
471
473
474
476
478
481
485
487
493
496
501
506
508
513
520
523
527
532
542
553
553
555
557
11.4 Generating Functions
559
11.5 Some Results on Partitions
563
11.6 Graphical Methods
565
11.7 Congruence Properties of Partitions
569
Exercises
573
12 Bailey Chains
577
12.1 Rogers\'s Second Proof of the Rogers-Ramanujan Identities
577
12.2 Bailey\'s Lemma
582
586
589
590
595
595
597
B
Summability and Fractional Integration
599
599
602
604
605
607
C
Asymptotic Expansions
611
C. 1
Asymptotic Expansion
611
C.2
Properties of Asymptotic Expansions
612
C.3
Watson\'s Lemma
614
C .4
The Ratio of Two Gamma Functions
615
Exercises
616
D
Euler-Maclaurin Summation Formula
617
D.I
Introduction
617
D.2
The Euler-Maclaurin Formula
619
D.3
Applications
621
D.4
The Poisson Summation Formula
623
Exercises
627
E
Lagrange Inversion Formula
629
E. 1
Reversion of Series
629
E.2
A Basic Lemma
630
E.3
Lambert\'s Identity
631
E.4
Whipple\'s Transformation
632
Exercises
634
F
Series Solutions of Differential Equations
637
F.I
Ordinary Points
637
F.2 Singular Points
638
F.3 Regular Singular Points
639
Bibliography
641
Index
655
Subject Index
659
Symbol Index
661