Q-factorial

The q-factorial is the q-analog of the factorial function.Wolfram Mathworld It is written \([n]_q!\) or \(\mathrm{faq}(n,q)\) and is defined as
\([n]_q! = \prod^{n - 1}_{i = 0} \left(\textstyle\sum^{i}_{j = 0} q^j\right) = q^0 \cdot \left(q^0 + q^1\right) \cdot \left(q^0 + q^1 + q^2\right) \cdot \ldots \cdot \left(q^0 + q^1 + \ldots + q^{n - 1}\right)\)
As with all q-analogs, letting \(q = 1\) produces the ordinary factorial.
Based on the q-factorial, we can define the q-exponential function:
\(e^x_q = \sum_{i = 0}^{\infty} \frac{x^i}{[i]_q!} = \frac{1}{[0]_q!} + \frac{x}{[1]_q!} + \frac{x^2}{[2]_q!} + \frac{x^3}{[3]_q!} + \cdots\)
as well as q-trigonometric functions \(\sin_q x = \frac{e^{ix}_q - e^{-ix}_q}{2i}\), \(\cos_q x = \frac{e^x_q + e^{-x}_q}{2}\), etc.
== Values ==
 	1 	2 	3 	4
1	1 	1 	1 	1
2	2 	3 	4 	5
3	6 	21 	52 	105
4	24 	315 	2,080 	8,925
5	120	9,765	251,680	3,043,425
== Sources ==
== See also ==
Category:Functions
Category:Factorials