Difference between revisions of "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 Dixon 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