*(English)*Zbl 0986.05001

The book is an elementary introduction to the two types of quantum calculus, $h$-calculus (that is the calculus of finite differences) and $q$-calculus. The main emphasis is on $q$-calculus. The authors define and study the $q$-derivative and $q$-antiderivative, the Jackson integral, $q$-analogs of classical objects of combinatorics, like binomial coefficients, etc., analogs of elementary and special functions (trigonometric, exponential, hypergeometric, gamma and beta functions).

The usefulness of $q$-analysis for classical problems of combinatorics and number theory is illustrated by proofs of the explicit formulas of Gauss and Jacobi for the number of partitions of an integer into a sum of two and of four squares.

Within $h$-calculus, the authors discuss the Bernoulli numbers and polynomials, and the Euler-Maclaurin formula.

The title “Quantum calculus” can be seen as a hint to connections with quantum groups and their applications in mathematical physics. However the book does not treat these subjects remaining within classical analysis and combinatorics.

##### MSC:

05-01 | Textbooks (combinatorics) |

05A30 | $q$-calculus and related topics |

33-01 | Textbooks (special functions) |

11B65 | Binomial coefficients, etc. |

11B68 | Bernoulli and Euler numbers and polynomials |

33D05 | $q$-gamma functions, $q$-beta functions and integrals |

05A17 | Partitions of integers (combinatorics) |

33D15 | Basic hypergeometric functions of one variable, ${}_{r}{\phi}_{s}$ |

11-01 | Textbooks (number theory) |