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Quantum calculus. (English) Zbl 0986.05001
Universitext. New York, NY: Springer. ix, 112 p. (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 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to combinatorics 05A30 $$q$$-calculus and related topics 33-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to special functions 11B65 Binomial coefficients; factorials; $$q$$-identities 11B68 Bernoulli and Euler numbers and polynomials 33D05 $$q$$-gamma functions, $$q$$-beta functions and integrals 05A17 Combinatorial aspects of partitions of integers 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$ 11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory