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

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

6.2 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

6.9 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

7.3

7.4

7.5

7.6

7.7

7.8

7.9

7.10

7.11

7.12

7.13

7.14

7.15

7.16

7.17

7.18

7.19

7.20

7.21

7.22

7.23

7.24