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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.

05-01Textbooks (combinatorics)
05A30q-calculus and related topics
33-01Textbooks (special functions)
11B65Binomial coefficients, etc.
11B68Bernoulli and Euler numbers and polynomials
33D05q-gamma functions, q-beta functions and integrals
05A17Partitions of integers (combinatorics)
33D15Basic hypergeometric functions of one variable, r φ s
11-01Textbooks (number theory)