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Quantum calculus. (English) Zbl 0986.05001
Universitext. New York, NY: Springer. ix, 112 p. EUR 34.95; sFr. 58.00; £24.50; \$ 29.95 (2002).

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)