Quantum calculus. (English) Zbl 0986.05001

Kac, Victor; Cheung, Pokman

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.

Reviewer: Anatoly N.Kochubei (Kyïv)

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

Keywords:

\(q\)-derivative; \(q\)-antiderivative; Bernoulli polynomial; Bernoulli number; partition; \(q\)-hypergeometric function; \(q\)-beta function; \(q\)-gamma function