High Quality Content by WIKIPEDIA articles! In mathematics, in the area of combinatorics and special functions, a q-analog is, roughly speaking, a theorem or identity in the variable q 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.

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