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Zeta and q-Zeta Functions and Associated Series and Integrals

by: Srivastava, H. M.; Choi, Junesang

Zeta and q-Zeta Functions and Associated Series and Integrals

by: Srivastava, H. M.; Choi, Junesang

ISBN 13:
9780123852182
ISBN 10:
0123852188
Format: Hardcover
Copyright: 10/25/2011
Publisher: Elsevier Science Ltd

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Summary

Zeta and q-Zeta Functions and Associated Series and Integrals is a thoroughly revised, enlarged and updated version of Series Associated with the Zeta and Related Functions. Many of the chapters and sections of the book have been significantly modified or rewritten and a new chapter on the theory and applications of the basic (or q-) extensions of various Special Functions is included. This book will be invaluable as it covers not only detailed and systematic presentations of the theory and applications of the various methods and techniques used in dealing with many different classes of series and integrals associated with the Zeta and related functions but stimulating historical accounts of a large number of problems and well-classified tables of series and integrals. Detailed and systematic presentations of the theory and applications of the various methods and techniques used in dealing with many different classes of series and integrals associated with the Zeta and related functions

Preface	p. xi
Acknowledgements	p. xv
Introduction and Preliminaries	p. 1
Gamma and Beta Functions	p. 1
The Gamma Function	p. 1
Pochhammer's Symbol and the Factorial Function	p. 4
Multiplication Formulas of Legendre and Gauss	p. 6
Stirling's Formula for n! and its Generalizations	p. 6
The Beta Function	p. 7
The Incomplete Gamma Functions	p. 10
The Incomplete Beta Functions	p. 10
The Error Functions	p. 11
The Bohr-Mollerup Theorem	p. 12
The Euler-Mascheroni Constant ¿	p. 13
A Set of Known Integral Representations for ¿	p. 15
Further Integral Representations for ¿	p. 18
From an Application of the Residue Calculus	p. 22
Polygamma Functions	p. 24
The Psi (or Digamma) Function	p. 24
Integral Representations for ¿(z)	p. 25
Gauss's Formulas for ¿(p-q)	p. 30
Special Values of ¿(z)	p. 31
The Polygamma Functions	p. 33
Special Values of ¿⁽ⁿ⁾(z)	p. 34
The Asymptotic Expansion for ¿(z)	p. 36
The Multiple Gamma Functions	p. 38
The Double Gamma Function ¿₂	p. 38
Integral Formulas Involving the Double Gamma Function	p. 45
The Evaluation of an Integral Involving log G(z)	p. 52
The Multiple Gamma Functions	p. 56
The Triple Gamma Function ¿₃	p. 58
The Gaussian Hypergeometric Function and its Generalization	p. 63
The Gauss Hypergeometric Equation	p. 63
Gauss's Hypergeometric Series	p. 64
The Hypergeometric Series and Its Analytic Continuation	p. 65
Linear, Quadratic and Cubic Transformations	p. 67
Hypergeometric Representations of Elementary Functions	p. 67
Hypergeometric Representations of Other Functions	p. 68
The Confluent Hypergeometric Function	p. 69
Important Properties of Kummer's Confluent Hypergeometric Function	p. 70
The Generalized (Gauss and Kummer) Hypergeometric Function	p. 71
Analytic Continuation of the Generalized Hypergeometric Function	p. 72
Functions Expressible in Terms of the _pF_q Function	p. 73
Stirling Numbers of the First and Second Kind	p. 76
Stirling Numbers of the First Kind	p. 76
Stirling Numbers of the Second Kind	p. 78
Relationships Among Stirling Numbers of the First and Second Kind and Bernoulli Numbers	p. 79
Bernoulli, Euler and Genocchi Polynomials and Numbers	p. 81
Bernoulli Polynomials and Numbers	p. 81
The Generalized Bernoulli Polynomials and Numbers	p. 83
Euler Polynomials and Numbers	p. 86
Fourier Series Expansions of Bernoulli and Euler Polynomials	p. 87
Relations Between Bernoulli and Euler Polynomials	p. 88
The Generalized Euler Polynomials and Numbers	p. 88
Genocchi Polynomials and Numbers	p. 90
Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi Polynomials and Numbers	p. 91
Apostol-Bernoulli Polynomials and Numbers	p. 91
Apostol-Genocchi Polynomials and Numbers	p. 98
Important Remarks and Observations	p. 99
Generalizations and Unified Presentations of the Apostol Type Polynomials	p. 100
Inequalities for the Gamma Function and the Double Gamma Function	p. 105
The Gamma Function and Its Relatives	p. 105
The Double Gamma Function	p. 112
Problems	p. 112
The Zeta and Related Functions	p. 141
Multiple Hurwitz Zeta Functions	p. 141
The Analytic Continuation of ¿_n (s, a)	p. 142
Relationship between ¿_n (s, x) and B^(¿)_n (x)	p. 150
The Vardi-Barnes Multiple Gamma Functions	p. 153
The Hurwitz (or Generalized) Zeta Function	p. 155
Hurwitz's Formula for ¿(s, a)	p. 156
Hermite's Formula for ¿(s, a)	p. 157
Further Integral Representations for ¿ (s, a)	p. 159
Some Applications of the Derivative Formula (17)	p. 160
Another Form for ¿₂(a)	p. 162
The Riemann Zeta Function	p. 164
Riemann's Functional Equation for ¿(s)t	p. 166
Relationship between ¿ (s) and the Mathematical Constants B and C	p. 167
Integral Representations for ¿ (s)	p. 169
A Summation Identity for ¿ (n)	p. 172
Polylogarithm Functions	p. 175
The Dilogarithm Function	p. 176
Clausen's Integral (or Function)	p. 181
The Trilogarithm Function	p. 183
The Polylogarithm Functions	p. 185
The Log-Sine Integrals	p. 191
Hurwitz-Lerch Zeta Functions	p. 194
The Taylor Series Expansion of the Lipschitz-Lerch Transcendent L(x, s, a)	p. 198
Evaluation of L(x, ùn, a)	p. 199
Generalizations of the Hurwitz-Lerch Zeta Function	p. 200
Analytic Continuations of Multiple Zeta Functions	p. 213
Generalized Functions of Gel'fand and Shilov	p. 213
Euler-Maclaurin Summation Formula	p. 220
Problems	p. 224
Series Involving Zeta Functions	p. 245
Historical Introduction	p. 245
Use of the Binomial Theorem	p. 247
Applications of Theorems 3.1 and 3.2	p. 257
Use of Generating Functions	p. 261
Series Involving Polygamma Functions	p. 266
Series Involving Polylogarithm Functions	p. 267
Use of Multiple Gamma Functions	p. 269
Evaluation by Using the Gamma Function	p. 269
Evaluation in Terms of Catalan's Constant G	p. 339
Further Evaluation by Using the Triple Gamma Function	p. 344
Applications of Corollary 3.3	p. 348
Use of Hypergeometric Identities	p. 350
Series Derivable from Gauss's Summation Formula 1.4(7)	p. 351
Series Derivable from Kummer's Formula (3)	p. 354
Series Derivable from Other Hypergeometric Summation Formulas	p. 358
Further Summation Formulas Related to Generalized Harmonic Numbers	p. 361
Other Methods and their Applications	p. 364
The Weierstrass Canonical Product Form for the Gamma Function	p. 364
Evaluation by Using Infinite Products	p. 366
Higher-Order Derivatives of the Gamma Function	p. 369
Applications of Series Involving the Zeta Function	p. 375
The Multiple Gamma Functions	p. 375
Mathieu Series	p. 382
Problems	p. 389
Evaluations and Series Representations	p. 399
Evaluation of ¿ (2n)	p. 399
The General Case of ¿ (2n)	p. 402
Rapidly Convergent Series for ¿ (2n + 1)	p. 405
Remarks and Observations	p. 409
Further Series Representations	p. 415
Computational Results	p. 422
Problems	p. 433
Determinants of the Laplacians	p. 445
The H-Dimensional Problem	p. 445
Computations Using the Simple and Multiple Gamma Functions	p. 448
Factorizations Into Simple and Multiple Gamma Functions	p. 448
Evaluations of det' ¿_n (n = 1, 2, 3)	p. 452
Computations Using Series of Zeta Functions	p. 457
Computations using Zeta Regularized Products	p. 465
A Lemma on Zeta Regularized Products and a Main Theorem	p. 467
Computations for small n	p. 471
Remarks and Observations	p. 472
Problems	p. 473
q-Extensions of Some Special Functions and Polynomials	p. 479
q-Shifted Factorials and q-Binomial Coefficients	p. 479
q-Derivative, q-Antiderivative and Jackson q-Integral	p. 483
q-Derivative	p. 484
q-Antiderivative and Jackson q-Integral	p. 484
q-Binomial Theorem	p. 487
q-Gamma Function and q-Beta Function	p. 490
q-Gamma Function	p. 490
q/-Beta Function	p. 495
A q-Extension of the Multiple Gamma Functions	p. 497
q-Bernoulli Numbers and (/-Bernoulli Polynomials	p. 499
q-Stirlmg Numbers of the Second Kind	p. 504
The Polynomial ¿_k(x) = ¿_k;q(x)	p. 506
q-Euler Numbers and g-Euler Polynomials	p. 509
The q-Apostol-Bernoulli Polynomials B⁽ⁿ⁾_k (x, ¿) of Order n	p. 513
The q-Apostol-Euler Polynomials ¿⁽ⁿ⁾_k (x, ¿) of Order n	p. 518
A Generalized g-Zeta Function	p. 519
An Auxiliary Function Defining Generalized	p. 519
Application of Euler-Maclaurin Summation Formula	p. 524
	p. 530
Analytic Continuation of g_q and ¿_q	p. 530
Analytic Continuation of Multiple Zeta Functions	p. 533
Special Values of ¿_q (s1, s2)	p. 541
Problems	p. 542
Miscellaneous Results	p. 555
A Set of Useful Mathematical Constants	p. 555
Euler-Mascheroni Constant ¿	p. 555
Series Representations for ¿	p. 556
A Class of Constants Analogous to {D_k}	p. 560
Other Classes of Mathematical Constants	p. 563
Log-Sine Integrals Involving Series Associated with the Zeta Function and Polylogarithms	p. 568
Analogous Log-Sine Integrals	p. 571
Remarks on Cl_n (¿) and G1_n (¿)	p. 575
Further Remarks and Observations	p. 578
Applications of the Gamma and Polygamma Functions Involving Convolutions of the Rayleigh Functions	p. 581
Series Expressible in Terms of the ¿-Function	p. 582
Convolutions of the Rayleigh Functions	p. 584
Bernoulli and Euler Polynomials at Rational Arguments	p. 587
The Cvijovic-Klinowski Summation Formulas	p. 588
Srivastava's Shorter Proofs of Theorem 7.3 and Theorem 7.4	p. 589
Formulas Involving the Hurwitz-Lerch Zeta Function	p. 591
An Application of Lerch's Functional Equation 2.5(29)	p. 593
Closed-Form Summation of Trigonometric Series	p. 594
Problems	p. 597
Bibliography	p. 603
Table of Contents provided by Ingram. All Rights Reserved.

Amazon no longer offers textbook rentals. We do!

Zeta and q-Zeta Functions and Associated Series and Integrals

Zeta and q-Zeta Functions and Associated Series and Integrals

9780123852182

0123852188

Summary

Table of Contents

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