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*}
ページ情報
| タイトル | 拡張多重階乗の簡単な値 |
| URL | https://www.nomuramath.com/zlx1htzs/ |
| SNSボタン |
負の多重階乗
\[
\left(-\left(qn+r\right)\right)!_{n}=\frac{\left(-1\right)^{q}}{\left(qn-\left(n-r\right)\right)!_{n}}
\]
ウォリス積分の拡張2重階乗表示
\[
\int_{0}^{\frac{\pi}{2}}\sin^{n}\theta d\theta=\frac{\left(n-1\right)!^{2}}{\left(n\right)!^{2}}\sqrt{\frac{\pi}{2}}
\]
多重階乗と拡張多重階乗の定義
\[
\left(x\right)!^{n}=n^{\frac{x-1}{n}}\frac{\left(\frac{x}{n}\right)!}{\left(\frac{1}{n}\right)!}
\]
多重階乗同士の関係
\[
\left(qn+r\right)!^{n}=r!^{n}\frac{\left(qn+r\right)!_{n}}{r!_{n}}
\]