큐-팩토리얼(q-factorial)은 포흐하머 심볼와 큐-넘버가 작동하는 큐-아날로그 팩토리얼(계승)이다.
큐-팩토리얼로부터의 가우스 계수, 또는 가우스 다항식으로 알려진 가우스 이항 계수
큐-다항계수는 다음과 같이 정의 할 수 있다.
![{\displaystyle [n]_{q}!={{(q;q)_{n}} \over {(1-q)^{n}}}\;\;\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3a48ed2b4c0924ec30253483d67158756b140ee)
![{\displaystyle [n]_{q}={{1-q^{n}} \over {1-q}}=q^{0}+q^{1}+q^{2}+\ldots +q^{n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48e3f2e7f13e6db4d3c9b2eda6ead6e3b2ae9527)
![{\displaystyle [n]_{q}={{1-q^{n}} \over {1-q}}=1+q+q^{2}+\ldots +q^{n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/550c312c2671b1dea7dacfb0e0ff2047d08f3257)
![{\displaystyle [n]_{q}!=\prod _{k=1}^{n}[k]_{q}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1d414de22587bf4651c639972c5bee5d88e18b4)
![{\displaystyle \;\;\;\;=[1]_{q}\cdot [2]_{q}\cdot [3]_{q}\cdots [n-1]_{q}\cdot [n]_{q}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5006916a6d95e9f506488a8462a7d15f9174640)
![{\displaystyle [n]_{q}!=\Gamma _{q}(n+1)=\prod _{k=1}^{n}[k]_{q}=[1]_{q}\cdot [2]_{q}\cdot [3]_{q}\cdots [n-1]_{q}\cdot [n]_{q}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d6514f70bd4f8a1fc506888e5dec0c22727bb88)
![{\displaystyle \therefore \;\;[n]_{1}!=[1]_{1}\cdot [2]_{1}\cdot [3]_{1}\cdots [n-1]_{1}\cdot [n]_{1}=(1)\cdot (2)\cdot (3)\cdots (n-1)\cdot (n)=n!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/105508bb00e0574ef712d0a6cb9ed0c6b6a8aa49)
![{\displaystyle \therefore \;\;[n]_{q}!={{[n]_{q}!} \over {[n-k]_{q}!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cb4e8cd5547dbad6539f5ba0e6558b26017cbe3)
![{\displaystyle \therefore \;\;[n]_{q}!={{[n]_{q}!} \over {[n-n]_{q}!}}={{[n]_{q}!} \over {[0]_{q}!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a39b7c3393de6eddd1abd32809f5ae6c8f521b89)
![{\displaystyle \therefore \;\;[0]_{q}!={{[n]_{q}!} \over {[n]_{q}!}}={1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc73cd1a6827416e5b6dda4d94fad8e30e49662)
![{\displaystyle [1]_{q}!=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfe0922ddd815b0bf26400c77ff1f74b0c36870f)
![{\displaystyle [2]_{q}!=1+q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/109eebc1a42e40ff4340f74b3fb3bfe005356708)
![{\displaystyle [3]_{q}!=(1+q)(1+q+q^{2})=1+2q+2q^{2}+q^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0461ae8d05d251f310561599ba69d300a3ecf863)
![{\displaystyle [4]_{q}!=(1+q)(1+q+q^{2})(1+q+q^{2}+q^{3})=1+3q+5q^{2}+6q^{3}+5q^{4}+3q^{5}+q^{6}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e1cf27cf268a755f78302e2ccd8bb5e033d566)
![{\displaystyle {\binom {n}{k}}_{q}={{[n]_{q}!} \over {[k]_{q}!\;[n-k]_{q}!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95851b9cb96ae90409254c0e94f9bf9f45f52b66)
![{\displaystyle {\begin{bmatrix}n\\k_{1},\ldots ,k_{m}\end{bmatrix}}_{q}={\frac {[n]_{q}!}{[k_{1}]_{q}!\cdots [k_{m}]_{q}!}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92da1d15a7e7e891d587bdcb775556fcc9f7312e)
![{\displaystyle e_{q}^{x}=\sum _{n=0}^{\infty }{{x^{n}} \over {[n]_{q}!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69fab5fcb54e01509952de3560adb8c64bb2a66b)
