{{DISPLAYTITLE:List of ''q''-analogs This is a list of ''q''-analogs in

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and related fields.

Algebra

Iwahori–Hecke algebra In mathematics, the Iwahori–Hecke algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a deformation of the group algebra of a Coxeter group. Hecke algebras are quotients of the group rings of Artin braid groups. This ...

Quantum affine algebra In mathematics, a quantum affine algebra (or affine quantum group) is a Hopf algebra that is a ''q''-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independently by and as a special case of their g ...

Quantum enveloping algebra In mathematics, a quantum or quantized enveloping algebra is a ''q''-analog of a universal enveloping algebra. Given a Lie algebra \mathfrak, the quantum enveloping algebra is typically denoted as U_q(\mathfrak). The notation was introduced by Drin ...

Quantum group In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras) ...

Analysis

Jackson integral In q-analog theory, the Jackson integral series in the theory of special functions that expresses the operation inverse to q-differentiation. The Jackson integral was introduced by Frank Hilton Jackson. For methods of numerical evaluation, see an ...

* ''q''-derivative * ''q''-difference polynomial *

Quantum calculus Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. It defines "q-calculus" and "h-calculus", where h ostensibly stands for Planck's constant while ''q'' stan ...

Combinatorics

LLT polynomial In mathematics, an LLT polynomial is one of a family of symmetric functions introduced by Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon (1997) as ''q''-analogues of products of Schur functions. J. Haglund, M. Haiman, N. Loehr (2005) show ...

* ''q''-binomial coefficient * ''q''-Pochhammer symbol * ''q''-Vandermonde identity

Orthogonal polynomials

* ''q''-Bessel polynomials * ''q''-Charlier polynomials * ''q''-Hahn polynomials * ''q''-Jacobi polynomials: ** Big ''q''-Jacobi polynomials ** Continuous ''q''-Jacobi polynomials ** Little ''q''-Jacobi polynomials * ''q''-Krawtchouk polynomials * ''q''-Laguerre polynomials * ''q''-Meixner polynomials * ''q''-Meixner–Pollaczek polynomials * ''q''-Racah polynomials

Probability and statistics

* Gaussian ''q''-distribution * ''q''-exponential distribution * ''q''-Weibull diribution * Tsallis ''q''-Gaussian *

Tsallis entropy In physics, the Tsallis entropy is a generalization of the standard Boltzmann–Gibbs entropy. Overview The concept was introduced in 1988 by Constantino Tsallis as a basis for generalizing the standard statistical mechanics and is identical in f ...

Special functions

Basic hypergeometric series In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called h ...

Elliptic gamma function In mathematics, the elliptic gamma function is a generalization of the q-gamma function, which is itself the q-analog of the ordinary gamma function. It is closely related to a function studied by , and can be expressed in terms of the triple gamma ...