Q-analog

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 (from inside the complex unit circle in most situations). 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.

Introductory examples

Noticing that

:$lim\_\{q\; ightarrow\; 1\}frac\{1-q^n\}\{1-q\}=n$

(it is not necessary in finite expressions like this to restrict "q" to the inside of the unit circle), we define the "q"-analog of "n", also known as the "q"-bracket or "q"-number 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

:

Again, one recovers the usual factorial by taking the limit as $q\; ightarrow\; 1$.

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

:

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, "q" = "p"^"e", 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 :Letting "q" approach 1, we get the binomial coefficient: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.

ee also

* q-derivative
* Gaussian binomial
* q-theta function
* elliptic gamma function
* Stirling number
* Young tableau
* Jacobi theta function
* Modular form
* q-Vandermonde identity

References

* [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