Zeta and q-Zeta Functions and Associated Series and Integrals

Zeta and q-Zeta Functions and Associated Series and Integrals......Page 3
Copyright......Page 4
Preface......Page 5
Acknowledgements......Page 8
The Gamma Function......Page 10
Pochhammer\'s Symbol and the Factorial Function......Page 13
Stirling\'s Formula for n! and its Generalizations......Page 15
The Beta Function......Page 16
The Incomplete Beta Functions......Page 19
The Error Functions......Page 20
The Bohr-Mollerup Theorem......Page 21
1.2 The Euler-Mascheroni Constant γ......Page 22
A Set of Known Integral Representations for γ......Page 24
Further Integral Representations for γ......Page 27
From an Application of the Residue Calculus......Page 31
The Psi (or Digamma) Function......Page 33
Integral Representations for ψ(z)......Page 34
Gauss\'s Formulas for ψ(p/q)......Page 39
Special Values of ψ(z)......Page 40
The Polygamma Functions......Page 42
Special Values of ψ(n)(z)......Page 43
The Asymptotic Expansion for ψ(z)......Page 45
The Double Gamma Function Γ2......Page 47
Integral Formulas Involving the Double Gamma Function......Page 54
The Evaluation of an Integral Involving log G(z)......Page 61
The Multiple Gamma Functions......Page 65
The Triple Gamma Function Γ3......Page 67
A Multiplication Formula for the Γn......Page 70
The Gauss Hypergeometric Equation......Page 72
Gauss\'s Hypergeometric Series......Page 73
The Hypergeometric Series and Its Analytic Continuation......Page 74
Hypergeometric Representations of Elementary Functions......Page 76
Hypergeometric Representations of Other Functions......Page 77
The Confluent Hypergeometric Function......Page 78
Important Properties of Kummer\'s Confluent Hypergeometric Function......Page 79
The Generalized (Gauss and Kummer) Hypergeometric Function......Page 80
Analytic Continuation of the Generalized Hypergeometric Function......Page 81
Functions Expressible in Terms of the pFq Function......Page 82
Stirling Numbers of the First Kind......Page 85
Stirling Numbers of the Second Kind......Page 87
Relationships Among Stirling Numbers of the First and Second Kind and Bernoulli Numbers......Page 88
Bernoulli Polynomials and Numbers......Page 90
The Generalized Bernoulli Polynomials and Numbers......Page 92
Euler Polynomials and Numbers......Page 95
Fourier Series Expansions of Bernoulli and Euler Polynomials......Page 96
The Generalized Euler Polynomials and Numbers......Page 97
Genocchi Polynomials and Numbers......Page 99
Apostol-Bernoulli Polynomials and Numbers......Page 100
Apostol-Genocchi Polynomials and Numbers......Page 107
Important Remarks and Observations......Page 108
Generalizations and Unified Presentations of the Apostol Type Polynomials......Page 109
The Gamma Function and Its Relatives......Page 114
Problems......Page 121
2.1 Multiple Hurwitz Zeta Functions......Page 150
The Analytic Continuation of ζn (s,a)......Page 151
Relationship between n (s,x) and B n (α)(x)......Page 159
The Vardi-Barnes Multiple Gamma Functions......Page 162
2.2 The Hurwitz (or Generalized) Zeta Function......Page 164
Hurwitz\'s Formula for ζ(s,a)......Page 165
Hermite\'s Formula for ζ(s,a)......Page 166
Further Integral Representations for ζ(s,a)......Page 168
Some Applications of the Derivative Formula (17)......Page 169
Another Form for Γ2(a)......Page 171
2.3 The Riemann Zeta Function......Page 173
Riemann\'s Functional Equation for ζ(s)......Page 175
Relationship between ζ(s) and the Mathematical Constants B and C......Page 176
Integral Representations for ζ(s)......Page 178
A Summation Identity for ζ(n)......Page 181
2.4 Polylogarithm Functions......Page 184
The Dilogarithm Function......Page 185
Clausen\'s Integral (or Function)......Page 190
The Trilogarithm Function......Page 192
The Polylogarithm Functions......Page 194
The Log-Sine Integrals......Page 200
2.5 Hurwitz–Lerch Zeta Functions......Page 203
The Taylor Series Expansion of the Lipschitz-Lerch Transcendent L(x,s,a)......Page 207
Evaluation of L(x,–n,a)......Page 208
2.6 Generalizations of the Hurwitz–Lerch Zeta Function......Page 209
Generalized Functions of Gel\'fand and Shilov......Page 222
Euler-Maclaurin Summation Formula......Page 229
Problems......Page 233
3.1 Historical Introduction......Page 253
3.2 Use of the Binomial Theorem......Page 255
Applications of Theorems 3.1 and 3.2......Page 265
3.3 Use of Generating Functions......Page 269
Series Involving Polygamma Functions......Page 274
Series Involving Polylogarithm Functions......Page 275
Evaluation by Using the Gamma Function......Page 277
Evaluation in Terms of Catalan\'s Constant G......Page 347
Further Evaluation by Using the Triple Gamma Function......Page 352
Applications of Corollary 3.3......Page 356
3.5 Use of Hypergeometric Identities......Page 358
Series Derivable from Gauss\'s Summation Formula 1.4(7)......Page 359
Series Derivable from Kummer\'s Formula (3)......Page 362
Series Derivable from Other Hypergeometric Summation Formulas......Page 366
Further Summation Formulas Related to Generalized  Harmonic Numbers......Page 369
The Weierstrass Canonical Product Form for the Gamma Function......Page 372
Evaluation by Using Infinite Products......Page 374
Higher-Order Derivatives of the Gamma Function......Page 377
The Multiple Gamma Functions......Page 383
Mathieu Series......Page 390
Problems......Page 397
4.1 Evaluation of ζ(2n)......Page 406
The General Case of ζ(2n)......Page 409
4.2 Rapidly Convergent Series for ζ(2n+1)......Page 412
Remarks and Observations......Page 416
4.3 Further Series Representations......Page 422
4.4 Computational Results......Page 429
Problems......Page 440
5.1 The n-Dimensional Problem......Page 451
Factorizations Into Simple and Multiple Gamma Functions......Page 454
Evaluations of det\' Δn (n=1, 2, 3)......Page 458
5.3 Computations Using Series of Zeta Functions......Page 463
5.4 Computations using Zeta Regularized Products......Page 471
A Lemma on Zeta Regularized Products and a Main Theorem......Page 473
Computations for small n......Page 477
5.5 Remarks and Observations......Page 478
Problems......Page 479
6.1 q-Shifted Factorials and q-Binomial Coefficients......Page 484
6.2 q-Derivative, q-Antiderivative and Jackson q-Integral......Page 488
q-Antiderivative and Jackson q-Integral......Page 489
6.3 q-Binomial Theorem......Page 492
q-Gamma Function......Page 495
q-Beta Function......Page 500
6.5 A q-Extension of the Multiple Gamma Functions......Page 502
6.6 q-Bernoulli Numbers and q-Bernoulli Polynomials......Page 504
q-Stirling Numbers of the Second Kind......Page 509
The Polynomial βk(x)=βk;q(x)......Page 511
6.7 q-Euler Numbers and q-Euler Polynomials......Page 514
6.8 The q-Apostol-Bernoulli Polynomials Bk(n)(x;λ) of Order n......Page 518
6.9 The q-Apostol-Euler Polynomials Ek(n)(x;λ) of Order n......Page 523
An Auxiliary Function Defining Generalized q-Zeta Function......Page 524
Application of Euler-Maclaurin Summation Formula......Page 529
Analytic Continuation of gq and ζq......Page 535
Analytic Continuation of Multiple Zeta Functions......Page 538
Special Values of ζq (s1, s2 )......Page 546
Problems......Page 547
Euler-Mascheroni Constant γ......Page 559
Series Representations for γ......Page 560
A Class of Constants Analogous to {Dk}......Page 564
Other Classes of Mathematical Constants......Page 567
7.2 Log-Sine Integrals Involving Series Associated with the Zeta Function and Polylogarithms......Page 572
Analogous Log-Sine Integrals......Page 575
Remarks on Cln(θ) and Gln(θ)......Page 579
Further Remarks and Observations......Page 582
7.3 Applications of the Gamma and Polygamma Functions Involving Convolutions of the Rayleigh Functions......Page 585
Series Expressible in Terms of the ψ-Function......Page 586
Convolutions of the Rayleigh Functions......Page 588
7.4 Bernoulli and Euler Polynomials at Rational Arguments......Page 591
The Cvijović-Klinowski Summation Formulas......Page 592
Srivastava\'s Shorter Proofs of Theorem 7.3 and Theorem 7.4......Page 593
Formulas Involving the Hurwitz-Lerch Zeta Function......Page 595
An Application of Lerch\'s Functional Equation 2.5(29)......Page 597
7.5 Closed-Form Summation of Trigonometric Series......Page 598
Problems......Page 601
Bibliography......Page 606