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q-factorial

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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.

ValuesEdit

1 2 3 4
1 1 1 1 1
2 2 3 4 5
3 6 21 52 105
4 24 315 2080 8925
5 120 9765 251680 3043425

Sources Edit

  1. [1]

See also Edit

Main article: Factorial
Multifactorials: Double factorial · Multifactorial
Falling and rising: Falling factorial · Rising factorial
Other mathematical variants: Alternating factorial · Hyperfactorial · q-factorial · Roman factorial · Subfactorial · Weak factorial
Tetrational growth: Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial

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