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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 ...
, a ''q''-analog of a theorem, identity or expression is a generalization involving a new parameter ''q'' that returns the original theorem, identity or expression in the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
as . Typically, mathematicians are interested in ''q''-analogs that arise naturally, rather than in arbitrarily contriving ''q''-analogs of known results. The earliest ''q''-analog studied in detail is the
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 ...
, which was introduced in the 19th century.Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood, 1983, , , ''q''-analogues are most frequently studied in the mathematical fields of
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
and
special function Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...
s. In these settings, the limit is often formal, as is often discrete-valued (for example, it may represent a
prime power In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17 ...
). ''q''-analogs find applications in a number of areas, including the study of
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
s and multi-fractal measures, and expressions for the
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
of chaotic
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s. The relationship to fractals and dynamical systems results from the fact that many fractal patterns have the symmetries of
Fuchsian group In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of t ...
s in general (see, for example Indra's pearls and the
Apollonian gasket In mathematics, an Apollonian gasket or Apollonian net is a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. It is named after Greek ...
) and the
modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...
in particular. The connection passes through
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...
and
ergodic theory Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
, where the
elliptic integral In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
s and
modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
s play a prominent role; the ''q''-series themselves are closely related to elliptic integrals. ''q''-analogs also appear in the study of
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) ...
s and in ''q''-deformed
superalgebra In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. T ...
s. The connection here is similar, in that much of
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
is set in the language of
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s, resulting in connections to
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s, which in turn relate to ''q''-series.


"Classical" ''q''-theory

Classical ''q''-theory begins with the ''q''-analogs of the nonnegative integers. The equality :\lim_\frac=n suggests that we define the ''q''-analog of ''n'', also known as the ''q''-bracket or ''q''-number of ''n'', to be : q=\frac = 1 + q + q^2 + \ldots + q^. By itself, the choice of this particular ''q''-analog among the many possible options is unmotivated. However, it appears naturally in several contexts. For example, having decided to use 'n''sub>''q'' as the ''q''-analog of ''n'', one may define the ''q''-analog of the
factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...
, known as the ''q''-factorial, by : \begin \big q! & = q \cdot q \cdots -1q \cdot q \\ pt& =\frac \cdot \frac \cdots \frac \cdot \frac \\ pt& =1\cdot (1+q)\cdots (1+q+\cdots + q^) \cdot (1+q+\cdots + q^). \end This ''q''-analog appears naturally in several contexts. Notably, while ''n''! counts the number of
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
s of length ''n'', 'n''sub>''q''! counts permutations while keeping track of the number of inversions. That is, if inv(''w'') denotes the number of inversions of the permutation ''w'' and ''S''''n'' denotes the set of permutations of length ''n'', we have : \sum_ q^ = q ! . In particular, one recovers the usual factorial by taking the limit as q\rightarrow 1. The ''q''-factorial also has a concise definition in terms of the ''q''-Pochhammer symbol, a basic building-block of all ''q''-theories: : q!=\frac. From the ''q''-factorials, one can move on to define the ''q''-binomial coefficients, also known as Gaussian coefficients, Gaussian polynomials, or
Gaussian binomial coefficient In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or ''q''-binomial coefficients) are ''q''-analogs of the binomial coefficients. The Gaussian binomial coefficient, written as \binom nk ...
s: : \binom_q = \frac. The ''q''-exponential is defined as: :e_q(x) = \sum_^\infty \frac. ''q''-trigonometric functions, along with a ''q''-Fourier transform have been defined in this context.


Combinatorial ''q''-analogs

The Gaussian coefficients count subspaces of a finite
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
. Let ''q'' be the number of elements in a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
. (The number ''q'' is then a power of a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, , so using the letter ''q'' is especially appropriate.) Then the number of ''k''-dimensional subspaces of the ''n''-dimensional vector space over the ''q''-element field equals : \binom nk_q . Letting ''q'' approach 1, we get the binomial coefficient : \binom nk, or in other words, the number of ''k''-element subsets of an ''n''-element set. Thus, one can regard a finite vector space as a ''q''-generalization of a set, and the subspaces as the ''q''-generalization of the subsets of the set. This has been a fruitful point of view in finding interesting new theorems. For example, there are ''q''-analogs of
Sperner's theorem Sperner's theorem, in discrete mathematics, describes the largest possible families of finite sets none of which contain any other sets in the family. It is one of the central results in extremal set theory. It is named after Emanuel Sperner, who ...
and
Ramsey theory Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask a ...
.


Cyclic sieving

Let ''q'' = (''e''2''i''/''n'')''d'' be the ''d''-th power of a primitive ''n''-th root of unity. Let ''C'' be a cyclic group of order ''n'' generated by an element ''c''. Let ''X'' be the set of ''k''-element subsets of the ''n''-element set . The group ''C'' has a canonical action on ''X'' given by sending ''c'' to the
cyclic permutation In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, ma ...
(1, 2, ..., ''n''). Then the number of fixed points of ''c''''d'' on ''X'' is equal to : \binom nk_q .


''q'' → 1

Conversely, by letting ''q'' vary and seeing ''q''-analogs as deformations, one can consider the combinatorial case of as a limit of ''q''-analogs as (often one cannot simply let in the formulae, hence the need to take a limit). This can be formalized in the
field with one element In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French–English pun, Fun. The name ...
, which recovers combinatorics as linear algebra over the field with one element: for example,
Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections th ...
s are simple
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
s over the field with one element.


Applications in the physical sciences

''q''-analogs are often found in exact solutions of many-body problems. In such cases, the limit usually corresponds to relatively simple dynamics, e.g., without nonlinear interactions, while gives insight into the complex nonlinear regime with feedbacks. An example from atomic physics is the model of molecular condensate creation from an ultra cold fermionic atomic gas during a sweep of an external magnetic field through the
Feshbach resonance In physics, a Feshbach resonance can occur upon collision of two slow atoms, when they temporarily stick together forming an unstable compound with short lifetime (so-called resonance). It is a feature of many-body systems in which a bound state i ...
. This process is described by a model with a ''q''-deformed version of the SU(2) algebra of operators, and its solution is described by ''q''-deformed exponential and binomial distributions.


See also

* List of ''q''-analogs *
Stirling number In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in a purely algebraic setting in his book ''Methodus differentialis'' (1730). They were rediscov ...
*
Young tableau In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups a ...


References

* Andrews, G. E., Askey, R. A. & Roy, R. (1999), ''Special Functions'', Cambridge University Press, Cambridge. * Gasper, G. & Rahman, M. (2004), ''Basic Hypergeometric Series'', Cambridge University Press, . * Ismail, M. E. H. (2005), ''Classical and Quantum Orthogonal Polynomials in One Variable'', Cambridge University Press. * Koekoek, R. & Swarttouw, R. F. (1998), ''The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue'', 98-17, Delft University of Technology, Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics.


External links

*{{Springer, id=/U/u095050, title=Umbral calculus
''q''-analog
from
MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...

''q''-bracket
from
MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...

''q''-factorial
from
MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...

''q''-binomial coefficient
from
MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...
Combinatorics *