Book:Thomas Ernst/A Comprehensive Treatment of q-Calculus

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Thomas Ernst: A Comprehensive Treatment of q-Calculus

Published $2012$, Springer Basel.


Contents

Introduction
1.1 A survey of the chapters
1.2 What is $q$-calculus?
1.2.1 Elementary series manipulation
1.3 Update on $q$-calculus
1.3.1 Current textbooks on this subject
1.3.2 Comparison with complex analysis
1.4 Comparison with nonstandard analysis
1.5 Comparison with the units of physics
1.6 Analogies between $q$-analysis and analysis
1.7 The first $q$-functions
2 The different languages of $q$-calculus
2.1 Schools --- traditions
2.2 Ramifications and minor Schools
2.2.1 Different notations
2.3 Finite differences and Bernoulli numbers
2.4 Umbral calculus, interpolation theory
2.5 Elliptic and Theta Schools and notations, the oldest roots -- the $q$-forerunners
2.6 Trigonometry, prosthaphaeresis, logarithms
2.7 The development of calculus
2.8 The Faulhaber mathematics
2.9 Descartes, Leibniz, Hindenburg, Arbogast
2.10 The Fakultäten
2.11 Königsberg School
2.12 Viennese School
2.13 Göttingen School
2.14 The combinatorial School: Gudermann, Grünert
2.15 Heidelberg School
2.16 Weierstraß, formal power series and the $\Gamma$ function
2.17 Halle $q$-analysis School
2.18 Jakob Friedrich Fries, Martin Ohm, Babbage, Peacock and Herschel
2.19 Different styles in $q$-analysis
3 Pre $q$-Analysis
3.1 The early connection between analytic number theory and $q$-series
3.2 Some aspects of combinatorial identities
3.2.1 Faà di Bruno formula
3.3 The duality between Bernoulli and Stirling numbers
3.4 Tangent numbers, Euler numbers
3.5 The occurrence of binomial coefficient identities in the literature
3.6 Nineteenth century: Catalna, Grigoriew, Imchenetsky
3.7 A short history of hypergeometric series
3.7.1 The $\Gamma$ function
3.7.2 Balanced and well-poised hypergeometric series
3.7.3 Fractional differentiation
3.7.4 Newton, Taylor, Stirling, Montmort
3.7.5 Euler's contribution
3.7.6 Vandermonde and Pfaffian summation formulas
3.7.7 Conic sections in the seventeenth century
3.7.8 The infinity in England
3.7.9 The infinity in the hands of Euler
3.7.10 The infinity, the binomial coefficients
3.7.11 Gauß' contribution
3.7.12 After Gauß: Clausen, Jacobi
3.7.13 Kummer's contribution
3.7.14 Cauchy, Riemann, Heine, Thomae, Papperitz
3.7.15 1800-1914; Sonine, Goursat, Stieltjes, Schafheitlin, Pochhammer, Mellin
3.7.16 First half of the twentieth century; England, USA
3.7.17 Special functions defined by integrals
3.7.18 Second half of the twentieth century
3.8 The Jacobi theta functions; different notations; properties
3.9 Meromorphic continuation and Riemann surfaces
3.10 Wave equation
3.11 Orthogonal polynomials
3.11.1 Legendre-d'Allonville-Murphy polynomials
3.11.2 Laguerre-Abel-Sonine-Murphy-Chebyshev-Halphen-Szegő polynomials
3.11.3 Jacobi polynomials
3.11.4 Hermite polynomials
4 The $q$-umbral calculus and semigroups. The Nørlund calculus of finite differences
4.1 The $q$-umbral calculus and semigroups
4.2 Finite differences
4.3 $q$-Appell polynomials
4.3.1 The generalized $q$-Bernoulli polynomials
4.3.2 The Ward $q$-Bernoulli numbers
4.3.3 The generalized JHC $q$-Bernoulli polynomials
4.3.4 NWA $q$-Euler polynomials
4.3.5 The NWA generalized $q$-Euler numbers
4.3.6 Several variables; $n$ negative
4.3.7 $q$-Euler-Maclaurin expansions
4.3.8 JHC polynomials of many variables; negative order
4.3.9 JHC $q$-Euler-Maclaurin expansions
4.3.10 Euler symbolic formula
4.3.11 Complementary argument theorems
4.4 $q$-Lucas and $q$-G polynomials
4.4.1 $q$-Lucas numbers
4.4.2 The $q$-G polynomials
4.4.3 Lucas and $G$ polynomials of negative order
4.4.4 Expansion formulas
4.5 The semiring of Ward numbers
5 $q$-Stirling numbers
5.1 Introduction
5.2 The Hahn-Cigler-Carlitz-Johnson approach
5.3 The Carlitz-Gould approach
5.4 The Jackson $q$-derivative as difference operator
5.5 Applications
6 The first $q$-functions
6.1 $q$-analogue, $q$-factorial, tilde operator
($6.1$)
($6.2$)
($6.3$)
(6.4)
(6.5)
($6.6$)
($6.7$)
(6.8)
(6.9)
(6.10)
(6.11)
(6.12)
(6.13)
(6.14)
(6.15)
(6.16)
(6.17)
(6.18)
(6.19)
(6.20)
6.2 The $q$-derivative
6.3 The $q$-integral
6.4 Two other tilde operators
6.5 The Gaussian $q$-binomial coefficients and the $q$-Leibniz theorem
6.5.1 Other formulas
6.6 Cigler's operational method for $q$-identities
6.7 Gould and Carlitz $q$-binomial coefficient identities
6.8 $q$-Exponential and $q$-trigonometric functions
(6.150)
(6.151)
(6.152)
(6.153)
(6.154)
(6.155)
(6.156)
(6.157)
(6.158)
(6.159)
(6.160)
(6.161)
(6.162)
(6.163)
(6.164)
(6.165)
(6.166)
(6.167)
(6.168)
(6.169)
(6.170)
(6.171)
(6.172)
(6.173)
(6.174)
(6.175)
(6.176)
(6.177)
(6.178)
(6.179)
(6.180)
6.9 The Heine function
6.10 Oscillations in $q$-analysis
6.11 The Jackson-Hahn-Cigler $q$-addition and $q$-analogues of the trigonometric functions
6.11.1 Further $q$-trigonometric functions
6.12 The Nalli-Ward-Al-Salam $q$-addition and some variants of the $q$-difference operator
6.13 Weierstraß elliptic functions and sigma functions
6.13.1 Elliptic functions
6.13.2 Connections with the $\Gamma_q$ function
6.14 The Chen-Liu operator or parameter augmentation
7 $q$-hypergeometric series
7.1 Definition of the $q$-hypergeometric series
7.1.1 $q$-difference equation for ${}_{p+1}\phi_p$
7.2 Balanced and well-poised $q$-hypergeometric series
7.3 Advantages of the Heine definition
7.4 $q$-Binomial theorem
7.5 Jacobi's elliptic functions expressed as real and imaginary parts of $q$-hypergeometric series with exponential argument (Heine)
7.6 The Jacobi triple product identity
7.7 $q$-contiguity relations
7.8 Heine $q$-transformations
7.8.1 The $q$-beta function
7.9 Heine's $q$-analogue of the Gauß summation formula
7.10 A $q$-analogue of the Pfaff-Saalschütz summation formula
7.11 Sears' ${}_4 \phi_3$ transformation
7.12 $q$-analogues of Thomae's transformations
7.13 The Bailey-Daum summation formula
7.14 A general expansion formula
7.15 A summation formula for a terminating very-well poised ${}_4 \phi_3$ series
7.16 A summation formula for a terminating very-well-poised ${}_6 \phi_5$ series
7.17 Watson's transformation formula for a terminating very-well-poised ${}_8 \phi_7$ series
7.18 Jackson's sum of a terminating very-well-poised balanced ${}_8 \phi_7$ series
7.18.1 Three corollaries
7.19 Watson's proof of the Rogers-Ramanujan identities
7.20 Bailey's 1929 transformation formula for a terminating, balanced, very-well-poised ${}_10 \phi_9$
7.21 Watson's $q$-analogue of the Barnes contour integral
7.22 Three $q$-analogues of the Euler integral formula for the function $\Gamma(x)$
7.23 Inequalities for the $\Gamma_q$ function
7.24 Summary of the umbral method
8 Sundry topics
8.1 Four $q$-summation formulas of Andrews
8.2 Some quadratic $q$-hypergeometric transformations
8.3 The Kummer ${}_2F_1(-1)$ formula and Jacobi's theta function
8.4 Another proof of the $q$-Dixon formula
8.5 A finite version of the $q$-Dixon formula
8.6 The Jackson summation formula for a finite, $2$-balanced, well-poised ${}_5 \phi_4$ series
8.7 The Jackson finite $q$-analogue of the Dixson formula
8.8 Other examples of $q$-special functions
8.9 $q$-analogues of two formulas by Brown and Eastham
8.10 The $q$-analogue of Truesdell's function
8.11 The Bailey transformation for $q$-series
8.12 $q$-Taylor formulas with remainder; the mean value theorem
8.12.1 The mean value theorem in $q$-analysis
8.13 Bilateral series
8.14 Fractional $q$-integrals
9 $q$-orthogonal polynomials
9.1 Ciglerian $q$-Laguerre polynomials
9.1.1 The different Laguerre-philosophies
9.1.2 The $q$-Laguerre polynomials
9.1.3 Generating functions and recurrences
9.1.4 Product expansions
9.1.5 Bilinear generating functions
9.1.6 Al-Salam operator expressions
9.1.7 The $q$-Laguerre Rodriguez operator
9.1.8 $q$-orthogonality
9.2 $q$-Jacobi polynomials
9.2.1 Definition and the Rodriguez formula
9.2.2 The $q$-Jacobi Rodriguez operator
9.2.3 More generating functions and recurrences
9.2.4 $q$-orthogonality
9.3 $q$-Legendre polynomials and Carlitz-Al-Salam polynomials
9.3.1 $q$-Legendre polynomials
9.3.2 Carlitz-Al-Salam polynomials
10 $q$-functions of several variables
10.1 The corresponding vector notation
10.2 Historical introduction
10.3 Transformations for basic double series
10.3.1 Double $q$-balanced series
10.3.2 Transformation formula of Carlitz-Srivastava
10.3.3 Three formulas of Andrews
10.3.4 $q$-Analogues of Carlson's formulas
10.4 The $q$-Appell function $\Phi_1$ as $q$-integral
10.5 $q$-analogues of some of Srivastava's formulas
10.6 Some $q$-formulas of Srivastava
10.6.1 Generating functions
10.6.2 Transformations
10.6.3 Double sum identities (Srivastava and Jain)
10.7 Two reduction formulas of Karlsson and Srivastava
10.8 $q$-analogues of reducibility theorems of Karlsson
10.9 $q$-Analogues of Burchnall-Chaundy expansions
10.9.1 $q$-analogues of Verma expansions
10.9.2 A similar formula
10.10 Multiple extensions of the Rothe-von Grüson-Gauß formula
10.11 An expansion formula in the spirit of Chaundy
10.12 Formulas according to Burchnall-Chaundy and Jackson
11 Linear partial $q$-difference equations
11.1 Introduction
11.2 Canoncial equations and symmetry techniques for $q$-series (Kalnins, Miller)
11.3 $q$-difference equations for $q$-Appell and $q$-Lauricella functions
12 $q$-Calculus and physics
12.1 The $q$-Coulomb problem and the $q$-hydrogen atom
12.2 Connections to knot theory
12.3 General relativity
12.4 Molecular and nuclear spectroscopy
12.5 Elementary particle physics and chemical physics
12.6 Electroweak interaction
12.7 String theory
12.8 Wess-Zumino model
12.9 Quantum Chromodynamics
References
Index before 1900
Index after 1900
Name index before 1900
Name index after 1900
Name index Physics
Notation index Chapter 1,2,6-9
Notation index Chapter 3
Notation index Chapter 4,5
Notation index Chapter 10-11
Notation index Chapter 12