q-Analog

A $q$-analog, also called a q-Extension or q-Generalization, is a mathematical expression parameterized by a quantity $q$ which generalizes a known expression and reduces to the known expression in the limit $q\to 1$. There are $q$-analogs of the Factorial, Binomial Coefficient, Derivative, Integral, Fibonacci Numbers, and so on. Koornwinder, Suslov, and Bustoz, have even managed some kind of $q$-Fourier analysis.

The $q$-analog of a mathematical object is generally called the ``$q$-object'', hence q-Binomial Coefficient, q-Factorial, etc. There are generally several $q$-analogs if there is one, and there is sometimes even a multibasic analog with independent $q_1$, $q_2$, ....

See also d-Analog, q-Beta Function, q-Binomial Coefficient, q-Binomial Theorem, q-Cosine, q-Derivative, q-Factorial, q-Gamma Function, q-Series, q-Sine, q-Vandermonde Sum

References

Exton, H. $q$-Hypergeometric Functions and Applications. New York: Halstead Press, 1983.