The earliest q-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.
[1] q-analogs are most frequently studied in the mathematical fields of combinatorics and special functions.
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.
The q-factorial also has a concise definition in terms of the q-Pochhammer symbol, a basic building-block of all q-theories: From the q-factorials, one can move on to define the q-binomial coefficients, also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients: The q-exponential is defined as: q-trigonometric functions, along with a q-Fourier transform, have been defined in this context.
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.
[citation needed] Let q = (e2πi/n)d be the d-th power of a primitive n-th root of unity.
Then the number of fixed points of cd on X is equal to Conversely, by letting q vary and seeing q-analogs as deformations, one can consider the combinatorial case of q = 1 as a limit of q-analogs as q → 1 (often one cannot simply let q = 1 in the formulae, hence the need to take a limit).