폴리감마 함수

감마 함수의 미분은 다음과 같이 폴리감마 함수(polygamma function) ψ n ( z ) {\displaystyle \psi _{n}(z)} 로 주어진다. ψ 0 ( z ) {\displaystyle \psi _{0}(z)} 은 ψ ( 0 ) ( z ) {\displaystyle \psi ^{(0)}(z)} 으로도 표기한다. 이것은 폴리감마 함수중 첫번째 함수인 디감마 함수이다. ψ n ( z ) = d n + 1 d z n + 1 ln ⁡ Γ ( z ) {\displaystyle {\psi _{n}(z)}={d^{n+1} \over {dz^{n+1}}}\ln \Gamma (z)} = d n d z n Γ ′ ( z ) Γ ( z ) {\displaystyle \;\;={d^{n} \over {dz^{n}}}{{\Gamma '(z)} \over {\Gamma (z)}}} = d n d z n ψ 0 ( z ) {\displaystyle \;\;={d^{n} \over {dz^{n}}}\psi _{0}(z)} Γ ( z ) = lim n → ∞ n z n ! ∏ k = 0 n ( z + k ) {\displaystyle \Gamma (z)=\lim _{n\to \infty }{\frac {n^{z}n!}{\displaystyle \prod _{k=0}^{n}(z+k)}}\!\;} Γ ( z ) = 1 z ∏ k = 1 ∞ ( 1 + 1 k ) z 1 + z k {\displaystyle \Gamma (z)={1 \over z}\prod _{k=1}^{\infty }{{\left(1+{1 \over k}\right)^{z}} \over {1+{z \over k}}}} ∴ Γ ′ ( z ) = ψ 0 ( z ) Γ ( z ) {\displaystyle \therefore \;\Gamma '(z)=\psi _{0}(z)\Gamma (z)} ψ 0 ( z ) = ψ ( z ) = Γ ′ ( z ) Γ ( z ) {\displaystyle \psi _{0}(z)=\psi (z)={{\Gamma '(z)} \over {\Gamma (z)}}} − Γ ′ ( 1 ) = γ = − ψ 0 ( 1 ) {\displaystyle -\Gamma '(1)=\gamma =-\psi _{0}(1)} γ = lim n → ∞ ( ∑ k = 1 n 1 k − ln ⁡ n ) = ∑ m = 2 ∞ ( − 1 ) m ζ ( m ) m ( γ {\displaystyle \gamma =\lim _{n\to \infty }\left(\sum _{k=1}^{n}{1 \over k}-\ln n\right)=\sum _{m=2}^{\infty }(-1)^{m}{{\zeta (m)} \over {m}}\;\;(\gamma \;} 오일러-마스케로니 상수 , ζ {\displaystyle ,\;\zeta \;} 리만 제타 함수 ) {\displaystyle )} m > 0 {\displaystyle m>0} 에서 감마 함수의 미분 Γ ′ ( m + 1 ) = m ! ⋅ ( − γ + ∑ k = 1 m 1 k ) {\displaystyle \Gamma '(m+1)=m!\cdot \left(-\gamma +\sum _{k=1}^{m}{{1} \over {k}}\right)} ∑ z = 1 ∞ ( 1 z − log e ⁡ ( z + 1 z ) ) = γ {\displaystyle \sum _{z=1}^{\infty }({\frac {1}{z}}-\log _{e}({\frac {z+1}{z}}))=\gamma } 디감마 함수는 감마 함수의 미분으로 정의된다. ψ n ( z ) {\displaystyle \psi _{n}(z)} 에서 n = 0 {\displaystyle n=0} 디감마 함수는 폴리감마 함수중 첫번째 함수로 주어진다. ψ 0 ( z ) {\displaystyle {\psi _{0}(z)}} = ψ ( z ) {\displaystyle ={\psi _{}(z)}} ψ n ( z ) {\displaystyle \psi _{n}(z)} n = 1 {\displaystyle n=1} ψ 1 ( z ) {\displaystyle {\psi _{1}(z)}} ψ 1 ( z ) = 1 z + 1 2 z 2 + ∑ k = 1 ∞ B 2 k z 2 k + 1 = ∑ k = 0 ∞ B k z k + 1 {\displaystyle \psi _{1}(z)={\frac {1}{z}}+{\frac {1}{2z^{2}}}+\sum _{k=1}^{\infty }{\frac {B_{2k}}{z^{2k+1}}}=\sum _{k=0}^{\infty }{\frac {B_{k}}{z^{k+1}}}} B {\displaystyle B} 는 베르누이 수 ψ ( n ) ( z ) = ( − 1 ) n + 1 n ! ∑ k = 0 ∞ 1 ( z + k ) n + 1 {\displaystyle \psi ^{(n)}(z)=(-1)^{n+1}\,n!\sum _{k=0}^{\infty }{{1} \over {(z+k)^{n+1}}}} ψ ( 0 ) ( z ) = ( − 1 ) 0 + 1 0 ! ∑ k = 0 ∞ 1 ( z + k ) 0 + 1 {\displaystyle \psi ^{(0)}(z)=(-1)^{0+1}\,0!\sum _{k=0}^{\infty }{{1} \over {(z+k)^{0+1}}}} ψ ( 0 ) ( z ) = − 1 ∑ k = 0 ∞ 1 ( z + k ) {\displaystyle \psi ^{(0)}(z)=-1\sum _{k=0}^{\infty }{{1} \over {(z+k)}}} ψ ( 0 ) ( 1 ) = − 1 ∑ k = 0 ∞ 1 ( 1 + k ) {\displaystyle \psi ^{(0)}(1)=-1\sum _{k=0}^{\infty }{{1} \over {(1+k)}}} Γ ( z ) = 1 z ∏ k = 1 ∞ ( 1 + 1 k ) z 1 + z k {\displaystyle \Gamma (z)={1 \over z}\prod _{k=1}^{\infty }{{\left(1+{1 \over k}\right)^{z}} \over {1+{z \over k}}}} Γ ( 1 ) = 1 1 ∏ k = 1 ∞ ( 1 + 1 1 ) 1 1 + 1 1 {\displaystyle \Gamma (1)={1 \over 1}\prod _{k=1}^{\infty }{{\left(1+{1 \over 1}\right)^{1}} \over {1+{1 \over 1}}}} Γ ( 1 ) = 1 {\displaystyle \Gamma (1)=1} ∴ Γ ′ ( z ) = ψ 0 ( z ) Γ ( z ) {\displaystyle \therefore \;\Gamma '(z)=\psi _{0}(z)\Gamma (z)} Γ ′ ( 1 ) = ψ 0 ( 1 ) Γ ( 1 ) {\displaystyle \;\Gamma '(1)=\psi _{0}(1)\Gamma (1)} Γ ′ ( 1 ) = ψ 0 ( 1 ) ⋅ 1 {\displaystyle \;\Gamma '(1)=\psi _{0}(1)\cdot 1} 후르비츠 제타 함수 표현 ψ ( n ) ( z ) = ( − 1 ) n + 1 n ! ζ ( n + 1 , z ) {\displaystyle \psi ^{(n)}(z)=(-1)^{n+1}\,n!\,\zeta (n+1,z)} 큐-폴리감마 함수는 폴리감마 함수가 큐-아날로그화 된것이다. ψ q ( n ) ( z ) = ∂ n ψ q ( z ) ∂ z n {\displaystyle \psi _{q}^{(n)}(z)={{\partial ^{n}\psi _{q}(z)} \over {\partial z^{n}}}} ψ q ( z ) = Γ q ′ ( z ) Γ q ( z ) , z > 0 {\displaystyle \psi _{q}(z)={{\Gamma '_{q}(z)} \over {\Gamma _{q}(z)}}\;\;\;,z>0} 큐 감마함수(q gamma function) f ( x + 1 ) = 1 − q z 1 − q f ( x ) {\displaystyle f(x+1)={{1-q^{z}} \over {1-q}}f(x)} q ∈ ( 0 , 1 ) {\displaystyle q\in \left(0,1\right)} 구간 예약 f ( 1 ) = 1 {\displaystyle f(1)=1} {\displaystyle } log ⁡ f ( x ) , x > 0 {\displaystyle \log f(x),x>0} f ( x ) = Γ q ( x ) {\displaystyle f(x)=\Gamma _{q}(x)} ∴ Γ q ( z ) = ( q ; q ) ∞ ( q z ; q ) ∞ ( 1 − q ) 1 − z {\displaystyle \therefore \Gamma _{q}(z)={{(q;q)\infty } \over {(q^{z};q)\infty }}(1-q)^{1-z}\;\;\;} 큐-포흐하머 기호 ( q ; q ) ∞ {\displaystyle \;(q;q)\infty } {\displaystyle } : {\displaystyle } : {\displaystyle } : {\displaystyle } 감마함수 람베르트 급수 에르되시-보와인 상수 큐-아날로그 디리클레 L-함수