| zeta函数,q-zeta函数,相伴级数与积分 |
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2020-06-21 00:00:00 |
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zeta函数,q-zeta函数,相伴级数与积分 内容简介
《zeta函数，q-zeta函数，相伴级数与积分》解析zeta函数，q—zeta函数，相伴级数与积分的新定义。《zeta函数，q-zeta函数，相伴级数与积分》对zeta函数和q—Zeta函数和相伴级数与积分做一次彻底修改，扩大和更新版本的系列版本。
zeta函数,q-zeta函数,相伴级数与积分 目录
Preface
Acknowledgements
1 Introduction and Preliminaries
1．1 Gamma and Beta Functions
The Gamma Function
Pochhammer's Symbol and the Factorial Function
Multiplication Formulas of Legendre and Gauss
Stirling's Formula for n！ and its Generalizations
The Beta Function
The Incomplete Gamma Functions
The Incomplete Beta Functions
The Error Functions
The Bohr—Mollerup Theorem
1．2 The Euler—Mascheroni Constant γ
A Set of Known Integral Representations for γ
Further Integral Representations for γ
From an Application of the Residue Calculus
1．3 Polygamma Functions
The Psi （or Digamma） Function
Integral Representation for ψ（z）
Gauss's Formulas for ψ（p／q）
Special Values of ψ（z）
The Polygamma Functions
Special Values of ψ（n） （z）
The Asymptotic Expansion for ψ（2）
1．4 The Multiple Gamma Functions
The Double Gamma Function F2
Integral Formulas Involving the Double Gamma Function
The Evaluation of an Integral Involving log G（z）
The Multiple Gamma Functions
The Triple Gamma Function г3
1．5 The Gaussian Hypergeometric Function and its Generalization
The Gauss Hypergeometric Equation
Gauss's Hypergeomeu：ic Series
The Hypergeometric Series and Its Analytic Continuation
Linear， Quadratic and Cubic Transformations
Hypergeometric Representations of Elementary Functions
Hypergeometric Representations of Other Functions
The Confluent Hypergeometric Function
Important Properties of Kummer's Confluent Hypergeometric Function
The Generalized （Gauss and Kummer） Hypergeometric Function
Analytic Continuation of the Generalized Hypergeometric Function
Functions Expressible in Terms of the pFq Function
1．6 Stirling Numbers of the First and Second Kind
Stirling Numbers of the First Kind
Stirling Numbers of the Second Kind
Relationships Among Stirling Numbers of the First and Second Kind and Bernoulli Numbers
1．7 Bernoulli， Euler and Genocchi Polynomials and Numbers
Bernoulli Polynomials and Numbers
The Generalized Bernoulli Polynomials and Numbers
Euler Polynomials and Numbers
Fourier Series Expansions of Bernoulli and Euler Polynomials
Relations Between Bernoulli and Euler Polynomials
The Generalized Euler Polynomials and Numbers
Genocchi Polynomials and Numbers
1．8 Apostol—Bernoulli， Apostol—Euler and Apostol—Genocchi Polynomials and Numbers
Apostol—Bernoulli Polynomials and Numbers
Apostol—Genocchi Polynomials and Numbers
Important Remarks and Observations
Generalizations and Unified Presentations of the Apostol Type Polynomials
1．9 Inequalities for the Gamma Function and the Double Gamma Function
The Gamma Function and Its Relatives
The Double Gamma Function
Problems
2 The Zeta and Related Functions
2．1 Multiple Hurwitz Zeta Functions
The Analytic Continuation of ζn（S， a）
Relationship between ζn（s．x） and B（a）n（x）
The Vardi—Barnes Multiple Gamma Functions
2．2 The Hurwitz （or Generalized） Zeta Function
Hurwitz's Formula for ζ（s， a）
Hermite's Formula for ζ（s， a）
Further Integral Representations for ζ（s， a）
Some Applications of the Derivative Formula （17）
Another Form for г2（a）
2．3 The Riemann Zeta Function
Riemann's Functional Equation for ζ（s）
Relationship between ζ（s） and the Mathematical
Constants B and C
Integral Representations for ζ（s）
A Summation Identity for ζ（n）
2．4 PolylogarithmFunctions
The Dilogarithm Function
Clausen's Integral （or Function）
The Trilogarithm Function
The Polylogarithm Functions
The Log—Sine Integrals
2．5 Hurwitz—Lerch Zeta Functions
The Taylor Series Expansion of the Lipschitz—Lerch
Transcendent L（x， s， a）
Evaluation of L（x， —n， a）
2．6 Generalizations of the Hurwitz—Lerch Zeta Function
2．7 Analytic Continuations of Multiple Zeta Functions
Generalized Functions of Gel'fand and Shilov
Euler—Maclaurin Summation Formula
Problems
3 Series Involving Zeta Functions
3．1 Historicallntroduction
3．2 Use of the Binomial Theorem
Applications of Theorems 3．1 and 3．2
3．3 Use of Generating Functions
Series Involving Polygamma Functions
Series Involving Polylogarithm Functions
3．4 Use of Multiple Gamma Functions
Evaluation by Using the Gamma Function
Evaluation in Terms of Catalan's Constant G
Further Evaluation by Using the Triple Gamma Function
Applications of Corollary 3．3
3．5 Use of Hypergeometric Identities
Series Derivable from Gauss's Summation Formula 1．4（7）
Series Derivable from Kummer's Formula （3）
Series Derivable from Other Hypergeometric Summation Formulas
Further Summation Formulas Related to Generalized Harmonic Numbers
3．6 Other Methods and their Applications
The Weierstrass Canonical Product Form for the Gamma Function
Evaluation by Using Infinite Products
Higher—Order Derivatives of the Gamma Function
3．7 Applications of Series Involving the Zeta Function
The Multiple Gamma Functions
Mathieu Series
Problems
4 Evaluations and Series Representations
4．1 Evaluation of ζ（2n）
The General Case of ζ（2n）
4．2 Rapidly Convergent Series for ζ（2n + 1）
Remarks and Observations
4．3 Further Series Representations
4．4 ComputationaIResults
Problems
5 Determinants of the Laplacians
5．1 The n—Dimensional Problem
5．2 Computations Using the Simple and Multiple Gamma Functions
Factorizations Into Simple and Multiple Gamma Functions
Evaluations of det'△n （n=1， 2， 3）
5．3 Computations Using Series of Zeta Functions
5．4 Computations using Zeta Regularized Products
A Lemma on Zeta Regularized Products and a Main Theorem
Computations for small n
5．5 Remarks and Observations
Problems
6 q—Extensions of Some Special Functions and Polynomials
6．1 q—Shifted Factorials and q—Binormal Coefficients
6．2 q—Derivative， q—Antiderivative and Jackson q—lntegral
q—Derivative
q—Antiderivative and Jackson q—lntegral
6．3 q—Binomial Theorem
6．4 q—Gamma Function and q—Beta Function
q—Gamma Function
q—Beta Function
6．5 A q—Extension of the Multiple Gamma Functions
6．6 q—Bernoulli Numbers and q—Bernoulli Polynomials
q—Stirling Numbers of the Second Kind
The Polynomial βk（x）=βk；q（X）
6．7 q—Euler Numbers and q—Euler Polynomials
6．8 The q—Apostol—Bernoulli Polynomials βk（n） （x；λ） of Order n
6．9 The q—Apostol—Euler Polynomials εEk（n）（x；λ） of Order n
6．10 A Generalized q—Zeta Function
An Auxiliary Function Defining Generalized q—Zeta Function
Application of Euler—Maclaurin Summation Formula
6．11 Multiple q—Zeta Functions
Analytic Continuation of gq and ζq
Analytic Continuation of Multiple Zeta Functions
Special Values of ζq （s1， s2）
Problems
7 Miscellaneous Results
7．1 A Set of Useful Mathematical Constants
Euler—Mascheroni Constant γ
Series Representations for γ
A Class of Constants Analogous to {Dk}
Other Classes of Mathematical Constants
7．2 Log—Sine Integrals Involving Series Associated with the Zeta Function and Polylogarithms
Analogous Log—Sine Integrals
Remarks on Cln （θ） and Gln （θ）
Further Remarks and Observations
7．3 Applications of the Gamma and Polygamma Functions Involving
Convolutions of the Rayleigh Functions
Series Expressible in Terms of the ψ—Function
Convolutions of the Rayleigh Functions
7．4 Bernoulli and Euler Polynomials at Rational Arguments
The Cvijovie—Klinowski Summation Formulas
Srivastava's Shorter Proofs of Theorem 7．3 and Theorem 7．4
Formulas Involving the Hurwitz—Lerch Zeta Function
An Application of Lerch's Functional Equation 2．5（29）
7．5 Closed—Form Summation of Trigonometric Series
Problems
Bibliography
编辑手记
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