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, -calculus (that is the calculus of finite differences) and -calculus. The main emphasis is on -calculus. The authors define and study the -derivative and -antiderivative, the Jackson integral, -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 -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 -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.