Book:Victor Kac/Quantum Calculus

Victor Kac and Pokman Cheung: Quantum Calculus

Published $2002$, Springer.

Contents

Introduction

1 $q$-Derivative and $h$-Derivative

$(1.1)$

$(1.2)$

$(1.3)$

$(1.4)$

$(1.5)$

$(1.6)$

$(1.7)$

$(1.8)$

$(1.9)$

$(1.10)$

$(1.11)$

$(1.12)$

$(1.13)$

$(1.14)$

$(1.15)$

2 Generalized Taylor's Formula for Polynomials

3 $q$-Analogue of $(x-a)^n$, $n$ an Integer, and $q$-Derivatives of Binomials

4 $q$-Taylor's Formula for Polynomials

5 Gauss's Binomial Formula and a Noncommutative Binomial Formula

6 Properties of $q$-Binomial Coefficients

7 $q$-Binomial Coefficients and Linear Algebra over Finite Fields

8 $q$-Taylor's Formula for Formal Power Series and Heine's Binomial Formula

9 Two Euler's Identities and two $q$-Exponential Functions

10 $q$-Trigonometric functions

11 Jacobi's Triple Product Identity

12 Classical Partition Function and Euler's Product Formula

13 $q$-Hypergeometric Functions and Heine's Formula

14 More on Heine's Formula and the General Binomial

15 Ramanujan Product Formula

16 Explicit Formulas for Sums of Two and of Four Squares

17 Explicit Formulas for Sums of Two of Four Triangular Numbers

18 $q$-Antiderivatives

19 Jackson Integral

20 Fundamental Theorem of $q$-Calculus and Integration by Parts

21 $q$-Gamma and $q$-Beta Functions

22 $h$-Derivative and $h$-Integral

23 Bernoulli Polynomials and Bernoulli Numbers

24 Sums of Powers

25 Euler-Maclaurin Formula

26 Symmetric Quantum Calculus

Appendix: A List of $q$-Antiderivatives

Literature

Index