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Zbl 0986.05001
Kac, Victor; Cheung, Pokman
Quantum calculus. (English)
[B] Universitext. New York, NY: Springer. ix, 112 p. EUR 34.95; sFr. 58.00; \sterling 24.50; \$ 29.95 (2002). ISBN 0-387-95341-8/pbk

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). \par 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. \par Within $h$-calculus, the authors discuss the Bernoulli numbers and polynomials, and the Euler-Maclaurin formula. \par 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.
[Anatoly N.Kochubei (Ky\" iv)]

MSC 2000:: *05-01 Textbooks (combinatorics)
05A30 q-calculus and related topics
33-01 Textbooks (special functions)
11B65 Binomial coefficients, etc.
11B68 Bernoulli numbers, etc.
33D05 q-gamma functions, q-beta functions and integrals
05A17 Partitions of integres (combinatorics)
33D15 Basic hypergeometric functions of one variable
11-01 Textbooks (number theory)

Keywords: $q$-derivative; $q$-antiderivative; Bernoulli polynomial; Bernoulli number; partition; $q$-hypergeometric function; $q$-beta function; $q$-gamma function

Cited in: Zbl 1250.11025 Zbl 1171.39006

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