zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.


MSC:
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)