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*}ページ情報
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ディガンマ関数・ポリガンマ関数の級数表示・テイラー展開と調和数・一般化調和数
\[
\psi\left(z\right)=-\gamma+H_{z-1}
\]
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
\[
\frac{x^{a}}{a!}=e^{x}\left(\frac{\Gamma\left(a+1,x\right)}{\Gamma\left(a+1\right)}-\frac{\Gamma\left(a,x\right)}{\Gamma\left(a\right)}\right)
\]
ガンマ関数の極限問題
\[
\lim_{x\rightarrow0}\frac{\Gamma(ax)}{\Gamma(x)}=\frac{1}{a}
\]