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{{Use American English|date = March 2019}}
{{DISPLAYTITLE:''q''-analog}}{{Short description|Type of mathematical generalization such that the original version is the limit as q approaches 1}}
In [[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 (mathematics)|limit]] as {{math|''q'' → 1}}.  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]], which was introduced in the 19th century.<ref>Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York:  Halstead Press, Chichester: Ellis Horwood, 1983, {{isbn|0853124914}} ,  {{isbn|0470274530}} , {{isbn|978-0470274538}}</ref>

''q''-analogues are most frequently studied in the mathematical fields of [[combinatorics]] and [[special function]]s.  In these settings, the limit {{math|''q'' → 1}} is often formal, as {{mvar|q}} is often discrete-valued (for example, it may represent a [[prime power]]).
''q''-analogs find applications in a number of areas, including the study of [[fractal]]s and [[multi-fractal measure]]s, and expressions for the [[entropy]] of chaotic [[dynamical system]]s. The relationship to fractals and dynamical systems results from the fact that many fractal patterns have the symmetries of [[Fuchsian group]]s in general (see, for example [[Indra's Pearls (book)|Indra's pearls]] and the [[Apollonian gasket]]) and the [[modular group]] in particular. The connection passes through [[hyperbolic geometry]] and [[ergodic theory]], where the [[elliptic integral]]s and [[modular form]]s play a prominent role; the [[q-series|''q''-series]] themselves are closely related to elliptic integrals.

''q''-analogs also appear in the study of [[quantum group]]s and in ''q''-deformed [[superalgebra]]s. The connection here is similar, in that much of [[string theory]] is set in the language of [[Riemann surface]]s, resulting in connections to [[elliptic curve]]s, which in turn relate to [[q-series|''q''-series]].

=="Classical" ''q''-theory==
Classical ''q''-theory begins with the ''q''-analogs of the nonnegative integers.<ref name="Ernst2003">{{cite journal |last1=Ernst |first1=Thomas |year=2003 |title=A Method for q-calculus |journal=Journal of Nonlinear Mathematical Physics |volume=10 |issue=4 |pages=487–525 |publisher= |doi=10.2991/jnmp.2003.10.4.5 |url=http://www.solnaschack.org/ernst/ernst_10-4cor.pdf|accessdate=2011-07-27|bibcode=2003JNMP...10..487E}}</ref>  The equality

:<math>\lim_{q\rightarrow 1}\frac{1-q^n}{1-q}=n</math>

suggests that we define the ''q''-analog of ''n'', also known as the '''''q''-bracket''' or '''''q''-number''' of ''n'', to be

:<math>[n]_q=\frac{1-q^n}{1-q} = 1 + q + q^2 + \ldots + q^{n - 1}.</math>

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''</sub> as the ''q''-analog of ''n'', one may define the ''q''-analog of the [[factorial]], known as the [[q-factorial|''q''-factorial]], by

: <math>
\begin{align}
\big[n]_q! & =[1]_q  \cdot [2]_q \cdots [n-1]_q  \cdot [n]_q \\[6pt]
& =\frac{1-q}{1-q} \cdot \frac{1-q^2}{1-q} \cdots \frac{1-q^{n-1}}{1-q} \cdot \frac{1-q^n}{1-q} \\[6pt]
& =1\cdot (1+q)\cdots (1+q+\cdots + q^{n-2}) \cdot (1+q+\cdots + q^{n-1}).
\end{align}
</math>

This ''q''-analog appears naturally in several contexts.  Notably, while ''n''! counts the number of [[permutation]]s of length ''n'', [''n'']<sub>''q''</sub>! counts permutations while keeping track of the number of [[inversion (discrete mathematics)|inversions]].  That is, if inv(''w'') denotes the number of inversions of the permutation ''w'' and ''S''<sub>''n''</sub> denotes the set of permutations of length ''n'', we have

:<math> \sum_{w \in S_n} q^{\text{inv}(w)} = [n]_q ! .</math>

In particular, one recovers the usual factorial by taking the limit as <math>q\rightarrow 1</math>.

The ''q''-factorial also has a concise definition in terms of the [[q-Pochhammer symbol|''q''-Pochhammer symbol]], a basic building-block of all ''q''-theories:

:<math>[n]_q!=\frac{(q;q)_n}{(1-q)^n}.</math>

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]]s:

:<math>
\binom{n}{k}_q
=
\frac{[n]_q!}{[n-k]_q! [k]_q!}.
</math>

The [[q-exponential|''q''-exponential]] is defined as:

:<math>e_q^x = \sum_{n=0}^\infty \frac{x^n}{[n]_q!}.</math>

''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]].  Let ''q'' be the number of elements in a [[finite field]].  (The number ''q'' is then a power of a [[prime number]], {{nowrap|1= ''q'' = ''p''<sup>''e''</sup>}}, 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 
:<math>
\binom nk_q .
</math>
Letting ''q'' approach 1, we get the binomial coefficient
:<math>
\binom nk,
</math>
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]] and [[Ramsey theory]].

== ''q'' → 1 ==
{{main|Field with one element}}
Conversely, by letting ''q'' vary and seeing ''q''-analogs as deformations, one can consider the combinatorial case of {{nowrap|1= ''q''&nbsp;=&nbsp;1}} as a limit of ''q''-analogs as {{nowrap|1= ''q''&nbsp;&rarr;&nbsp;1}} (often one cannot simply let {{nowrap|1= ''q''&nbsp;=&nbsp;1}} in the formulae, hence the need to take a limit).

This can be formalized in the [[field with one element]], which recovers combinatorics as linear algebra over the field with one element: for example, [[Weyl group]]s are simple [[algebraic group]]s over the field with one element.

==Applications in the physical sciences==
''q''-analogs are often found in exact solutions of many-body problems.{{cn|date=March 2018}} In such cases, the {{math|''q'' → 1}} limit usually corresponds to relatively simple dynamics, e.g., without nonlinear interactions, while {{math|''q'' < 1}} 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.<ref name='sun-16pra2'>{{cite journal|doi=10.1103/PhysRevA.94.033808|title=Landau-Zener extension of the Tavis-Cummings model: Structure of the solution|author1=C. Sun|author2=N. A. Sinitsyn |journal=[[Phys. Rev. A]]|volume=94|issue=3|year=2016|pages=033808|bibcode=2016PhRvA..94c3808S|arxiv=1606.08430}}</ref> 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|List of ''q''-analogs]]
* [[Stirling number]]
* [[Young tableau]]

==References==
*[[George Andrews (mathematician)|Andrews, G. E.]],  [[Richard Askey|Askey, R. A.]] & [[Ranjoy Roy|Roy, R.]] (1999), ''Special Functions'', Cambridge University Press, Cambridge.
* [[George Gasper|Gasper, G.]] & [[Mizan Rahman|Rahman, M.]]  (2004), ''Basic Hypergeometric Series'', Cambridge University Press, {{ISBN|0521833574}}.
* [[Mourad Ismail|Ismail, M. E. H.]] (2005), ''Classical and Quantum Orthogonal Polynomials in One Variable'',  Cambridge University Press.
*[[Roelof Koekoek|Koekoek, R.]]  & [[René F Swarttouw|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.
<references/>

==External links==
*{{Springer|id=/U/u095050|title=Umbral calculus}}
* [http://mathworld.wolfram.com/q-Analog.html ''q''-analog] from [[MathWorld]]
* [http://mathworld.wolfram.com/q-Bracket.html ''q''-bracket] from [[MathWorld]]
* [http://mathworld.wolfram.com/q-Factorial.html ''q''-factorial] from [[MathWorld]]
* [http://mathworld.wolfram.com/q-BinomialCoefficient.html ''q''-binomial coefficient] from [[MathWorld]]

[[Category:Combinatorics]]
[[Category:Q-analogs|*]]