큐-팩토리얼

큐-팩토리얼(q-factorial)은 포흐하머 심볼와 큐-넘버가 작동하는 큐-아날로그 팩토리얼(계승)이다. [ n ] q ! = ( q ; q ) n ( 1 − q ) n {\displaystyle [n]_{q}!={{(q;q)_{n}} \over {(1-q)^{n}}}\;\;\;} 여기서 ( q ; q ) n {\displaystyle \;(q;q)_{n}} 는 큐-포흐하머 기호 lim q → 1 1 − q n 1 − q = n {\displaystyle \lim _{q\rightarrow 1}{{1-q^{n}} \over {1-q}}=n} [ n ] q = 1 − q n 1 − q = q 0 + q 1 + q 2 + … + q n − 1 {\displaystyle [n]_{q}={{1-q^{n}} \over {1-q}}=q^{0}+q^{1}+q^{2}+\ldots +q^{n-1}} [ n ] q = 1 − q n 1 − q = 1 + q + q 2 + … + q n − 1 {\displaystyle [n]_{q}={{1-q^{n}} \over {1-q}}=1+q+q^{2}+\ldots +q^{n-1}} [ n ] q ! = ∏ k = 1 n [ k ] q {\displaystyle [n]_{q}!=\prod _{k=1}^{n}[k]_{q}} = [ 1 ] q ⋅ [ 2 ] q ⋅ [ 3 ] q ⋯ [ n − 1 ] q ⋅ [ n ] q {\displaystyle \;\;\;\;=[1]_{q}\cdot [2]_{q}\cdot [3]_{q}\cdots [n-1]_{q}\cdot [n]_{q}} = ( 1 − q 1 − q ) ⋅ ( 1 − q 2 1 − q ) ⋅ ( 1 − q 3 1 − q ) ⋯ ( 1 − q n − 1 1 − q ) ⋅ ( 1 − q n 1 − q ) {\displaystyle \;\;\;\;=\left({{1-q} \over {1-q}}\right)\cdot \left({{1-q^{2}} \over {1-q}}\right)\cdot \left({{1-q^{3}} \over {1-q}}\right)\cdots \left({{1-q^{n-1}} \over {1-q}}\right)\cdot \left({{1-q^{n}} \over {1-q}}\right)} = ( 1 ) ⋅ ( 1 + q ) ⋅ ( 1 + q + q 2 ) ⋯ ( 1 + q + ⋯ + q n − 2 ) ⋅ ( 1 + q + ⋯ + q n − 1 ) {\displaystyle \;\;\;=(1)\cdot (1+q)\cdot (1+q+q^{2})\cdots (1+q+\cdots +q^{n-2})\cdot (1+q+\cdots +q^{n-1})} = ( q ; q ) n ( 1 − q ) n {\displaystyle \;\;\;={{(q;q)_{n}} \over {(1-q)^{n}}}\;\;\;} [ n ] q ! = Γ q ( n + 1 ) = ∏ k = 1 n [ k ] q = [ 1 ] q ⋅ [ 2 ] q ⋅ [ 3 ] q ⋯ [ n − 1 ] q ⋅ [ n ] q {\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}} n ! = n ! ( n − k ) ! {\displaystyle n!={{n!} \over {(n-k)!}}} n ! = ( n − 0 ) ⋅ ( n − 1 ) ⋅ ( n − 2 ) ⋅ ⋅ ⋅ ⋅ ⋅ ( n − ( n − 1 ) + 1 ) ⋅ ( n − ( n − 0 ) + 1 ) {\displaystyle n!=(n-0)\cdot (n-1)\cdot (n-2)\cdot \cdot \cdot \cdot \cdot (n-(n-1)+1)\cdot (n-(n-0)+1)} n ! = ( n ) ⋅ ( n − 1 ) ⋅ ( n − 2 ) ⋅ ⋅ ⋅ ⋅ ⋅ ( n − ( n − 2 ) ) ⋅ ( n − ( n − 1 ) ) {\displaystyle n!=(n)\cdot (n-1)\cdot (n-2)\cdot \cdot \cdot \cdot \cdot (n-(n-2))\cdot (n-(n-1))} n ! = ( 1 ) ⋅ ( 2 ) ⋅ ( 3 ) ⋯ ( n − 1 ) ⋅ ( n ) {\displaystyle n!=(1)\cdot (2)\cdot (3)\cdots (n-1)\cdot (n)} q = 1 ( q → 1 ) {\displaystyle q=1(q\to 1)} ∴ [ n ] 1 ! = [ 1 ] 1 ⋅ [ 2 ] 1 ⋅ [ 3 ] 1 ⋯ [ n − 1 ] 1 ⋅ [ n ] 1 = ( 1 ) ⋅ ( 2 ) ⋅ ( 3 ) ⋯ ( n − 1 ) ⋅ ( n ) = n ! {\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!} ∴ [ n ] q ! = [ n ] q ! [ n − k ] q ! {\displaystyle \therefore \;\;[n]_{q}!={{[n]_{q}!} \over {[n-k]_{q}!}}} ∴ [ n ] q ! = [ n ] q ! [ n − n ] q ! = [ n ] q ! [ 0 ] q ! {\displaystyle \therefore \;\;[n]_{q}!={{[n]_{q}!} \over {[n-n]_{q}!}}={{[n]_{q}!} \over {[0]_{q}!}}} ∴ [ 0 ] q ! = [ n ] q ! [ n ] q ! = 1 {\displaystyle \therefore \;\;[0]_{q}!={{[n]_{q}!} \over {[n]_{q}!}}={1}} [ 1 ] q ! = 1 {\displaystyle [1]_{q}!=1} [ 2 ] q ! = 1 + q {\displaystyle [2]_{q}!=1+q} [ 3 ] q ! = ( 1 + q ) ( 1 + q + q 2 ) = 1 + 2 q + 2 q 2 + q 3 {\displaystyle [3]_{q}!=(1+q)(1+q+q^{2})=1+2q+2q^{2}+q^{3}} [ 4 ] q ! = ( 1 + q ) ( 1 + q + q 2 ) ( 1 + q + q 2 + q 3 ) = 1 + 3 q + 5 q 2 + 6 q 3 + 5 q 4 + 3 q 5 + q 6 {\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}} 큐-이항 계수(q-binomial coefficients) 큐-팩토리얼로부터의 가우스 계수, 또는 가우스 다항식으로 알려진 가우스 이항 계수 ( n k ) q = [ n ] q ! [ k ] q ! [ n − k ] q ! {\displaystyle {\binom {n}{k}}_{q}={{[n]_{q}!} \over {[k]_{q}!\;[n-k]_{q}!}}} 큐-다항계수는 다음과 같이 정의 할 수 있다. [ n k 1 , … , k m ] q = [ n ] q ! [ k 1 ] q ! ⋯ [ k m ] q ! , {\displaystyle {\begin{bmatrix}n\\k_{1},\ldots ,k_{m}\end{bmatrix}}_{q}={\frac {[n]_{q}!}{[k_{1}]_{q}!\cdots [k_{m}]_{q}!}},} 큐-지수 e {\displaystyle {\color {blue}{e}}} e q x = ∑ n = 0 ∞ x n [ n ] q ! {\displaystyle e_{q}^{x}=\sum _{n=0}^{\infty }{{x^{n}} \over {[n]_{q}!}}} 큐- s i n e {\displaystyle sine} sin q ⁡ ( x ) {\displaystyle \sin _{q}(x)} 큐-파이 π q {\displaystyle \pi _{q}} 큐-폴리감마 함수 큐-아날로그 초기하함수