The q-factorial is the q-analog of the factorial function.[1] 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[]
Falling and rising: Falling factorial · Rising factorial
Other mathematical variants: Alternating factorial · Hyperfactorial · q-factorial · Roman factorial · Subfactorial · Weak factorial · Primorial · Compositorial · Semiprimorial
Tetrational growth: Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover
Nested Factorials: Tetorial · Petorial · Ectorial · Zettorial · Yottorial
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Tetrofactorial · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial · Bouncing Factorial