Q-analog

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The correct title of this article is q-analog. The initial letter is shown capitalized due to technical restrictions.

In mathematics, in the area of combinatorics and special functions, a q-analog is, roughly speaking, a theorem or identity for a q-series that gives back a known result in the limit, as q → 1, inside the unit circle. The earliest q-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.

q-analogs find applications in a number of areas, including the study of fractals and multi-fractal measures, and expressions for the entropy of chaotic dynamical systems. The relationship to fractals and dynamical systems results from the fact that many fractal patterns have the symmetries of Fuchsian groups in general (see, for example 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 integrals and modular forms play a prominent role; the q-series themselves are closely related to elliptic integrals.

q-analogs also appear in the study of quantum groups and in q-deformed superalgebras. The connection here is similar, in that much of string theory is set in the language of Riemann surfaces, resulting in connections to elliptic curves, which in turn relate to q-series.

[edit] Examples

For convenience, the limit q → 1 inside the unit circle is written as the limit q → 1⁻, which suggests the limit through real values tending up to 1, which is in fact more restricted, though the difference is not usually significant.

Noticing that

$\lim_{q\rightarrow 1^-}\frac{1-q^n}{1-q}=n,$

we define the q-analog of n, also known as the q-bracket of n to be

$[n]_q=\frac{1-q^n}{1-q}.$

From this one can define the q-analog of the factorial, the q-factorial, as

$\big[n]_q!$	$=[1]_q [2]_q \cdots [n-1]_q [n]_q$
	$=\frac{1-q}{1-q} \frac{1-q^2}{1-q} \cdots \frac{1-q^{n-1}}{1-q} \frac{1-q^n}{1-q}$
	$=1(1+q)\cdots (1+q+\cdots + q^{n-2}) (1+q+\cdots + q^{n-1}).$

Again, one recovers the usual factorial by taking the limit as $q\rightarrow 1^{-}$ .

From the q-factorials, one can move on to define the q-binomial coefficients, also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomials:

$\begin{bmatrix} n\\ k \end{bmatrix}_q = \frac{[n]_q!}{[n-k]_q! [k]_q!}.$