拡張多重階乗の簡単な値
拡張多重階乗の簡単な値
(1)
\[ 0!^{n}=\frac{1}{\sqrt[n]{n}\left(\frac{1}{n}\right)!} \]
(2)
\[ 1!^{n}=1 \]
*
\(x!^{n}\)は拡張多重階乗。
(1)
\begin{align*} 0!^{n} & =n^{-\frac{1}{n}}\frac{0!}{\left(\frac{1}{n}\right)!}\cmt{\left(x\right)!^{n}=n^{\frac{x-1}{n}}\frac{\left(\frac{x}{n}\right)!}{\left(\frac{1}{n}\right)!}}\\ & =\frac{1}{\sqrt[n]{n}\left(\frac{1}{n}\right)!} \end{align*}
(2)
\begin{align*} 1!^{n} & =n^{0}\frac{\left(\frac{1}{n}\right)!}{\left(\frac{1}{n}\right)!}\\ & =1\cmt{n\ne0} \end{align*}
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2重階乗の逆数和
\[
\sum_{k=0}^{n}\frac{1}{\left(2k\right)!!}=\sqrt{e}\frac{\Gamma\left(n+1,\frac{1}{2}\right)}{\Gamma\left(n+1\right)}
\]
拡張多重階乗の漸化式
\[
x!^{n}=x\left(x-n\right)!^{n}
\]
多重階乗の階乗表示
\[
\left(qn+r\right)!_{n}=r!_{n}n^{q}\frac{\left(q+\frac{r}{n}\right)!}{\left(\frac{r}{n}\right)!}
\]
負の多重階乗
\[
\left(-\left(qn+r\right)\right)!_{n}=\frac{\left(-1\right)^{q}}{\left(qn-\left(n-r\right)\right)!_{n}}
\]