# Book:Victor Kac/Quantum Calculus

## Victor Kac and Pokman Cheung: Quantum Calculus

Published $2002$, Springer.

### Contents

Introduction
1 $q$-Derivative and $h$-Derivative
----------
$(1.5)$
----------
$(1.9)$
2 Generalized Taylor's Formula for Polynomials
3 $q$-Analogue of $(x-a)^n$, $n$ an Integer, and $q$-Derivatives of Binomials
$(3.1)$
----------
$(3.8)$
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
----------
$(9.5)$
----------
$(9.10)$
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