負の整数の階乗の商
負の整数の階乗の商
\(m,n\in\mathbb{Z}\)とする。
\begin{align*} \frac{\left(-m\right)!}{\left(-n\right)!} & =\left(-1\right)^{n-m}\frac{\Gamma\left(n\right)}{\Gamma\left(m\right)}\\ & =\left(-1\right)^{n-m}\frac{\left(n-1\right)!}{\left(m-1\right)!} \end{align*}
\(m,n\in\mathbb{Z}\)とする。
\begin{align*} \frac{\left(-m\right)!}{\left(-n\right)!} & =\left(-1\right)^{n-m}\frac{\Gamma\left(n\right)}{\Gamma\left(m\right)}\\ & =\left(-1\right)^{n-m}\frac{\left(n-1\right)!}{\left(m-1\right)!} \end{align*}
(0)
\begin{align*} \frac{\left(-m\right)!}{\left(-n\right)!} & =P\left(-m,-m+n\right)\\ & =\left(-1\right)^{n-m}Q\left(m,n-m\right)\\ & =\left(-1\right)^{n-m}P^{-1}\left(m-1,m-n\right)\\ & =\left(-1\right)^{n-m}\frac{\left(m-1-\left(m-n\right)\right)!}{\left(m-1\right)!}\\ & =\left(-1\right)^{n-m}\frac{\left(n-1\right)!}{\left(m-1\right)!}\\ & =\left(-1\right)^{n-m}\frac{\Gamma\left(n\right)}{\Gamma\left(m\right)} \end{align*}(0)-2
\begin{align*} \frac{\left(-m\right)!}{\left(-n\right)!} & =\lim_{\epsilon\rightarrow0}\frac{\left(-\left(m+\epsilon\right)\right)!}{\left(-\left(n+\epsilon\right)\right)!}\\ & =\lim_{\epsilon\rightarrow0}\frac{\Gamma\left(1-\left(m+\epsilon\right)\right)}{\Gamma\left(1-\left(n+\epsilon\right)\right)}\\ & =\lim_{\epsilon\rightarrow0}\frac{\Gamma\left(\left(m+\epsilon\right)\right)\pi\sin\left(\pi\left(n+\epsilon\right)\right)}{\Gamma\left(\left(n+\epsilon\right)\right)\pi\sin\left(\pi\left(m+\epsilon\right)\right)}\\ & =\lim_{\epsilon\rightarrow0}\frac{\Gamma\left(\left(m+\epsilon\right)\right)\pi\cos\left(\pi\left(n+\epsilon\right)\right)}{\Gamma\left(\left(n+\epsilon\right)\right)\pi\cos\left(\pi\left(m+\epsilon\right)\right)}\\ & =\frac{\Gamma\left(m\right)\cos\left(\pi n\right)}{\Gamma\left(n\right)\cos\left(\pi m\right)}\\ & =\left(-1\right)^{n-m}\frac{\Gamma\left(n\right)}{\Gamma\left(m\right)}\\ & =\left(-1\right)^{n-m}\frac{\left(n-1\right)!}{\left(m-1\right)!} \end{align*}
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そのままだとΓ(0)になる積分
\[
\int_{0}^{\infty}\left(x^{-1}e^{-x}-\frac{e^{-nx}}{1-e^{-x}}\right)dx=H_{n-1}-\gamma
\]
ポリガンマ関数同士の差の極限
\[
\lim_{z\rightarrow0}\left(\psi^{\left(n\right)}\left(z-m\right)-\psi^{\left(n\right)}\left(z\right)\right)=n!H_{m,n+1}
\]
階乗と階乗の逆数の母関数
\[
\frac{x^{a}}{a!}=e^{x}\left(\frac{\Gamma\left(a+1,x\right)}{\Gamma\left(a+1\right)}-\frac{\Gamma\left(a,x\right)}{\Gamma\left(a\right)}\right)
\]
第1種・第2種不完全ガンマ関数の整数値
\[
\gamma\left(n+1,x\right)=-e^{-x}\sum_{k=0}^{n}\left(P\left(n,k\right)x^{n-k}\right)+n!
\]