拡張多重階乗の漸化式
拡張多重階乗の漸化式
\[ x!^{n}=x\left(x-n\right)!^{n} \]
\[ x!^{n}=x\left(x-n\right)!^{n} \]
*
\(x!^{n}\)は拡張多重階乗。\begin{align*}
x!^{n} & =n^{\frac{x-1}{n}}\frac{\left(\frac{x}{n}\right)!}{\left(\frac{1}{n}\right)!}\\
& =n^{\frac{x-1}{n}}\frac{\frac{x}{n}\left(\frac{x}{n}-1\right)!}{\left(\frac{1}{n}\right)!}\\
& =xn^{\frac{\left(x-n\right)-1}{n}}\frac{\left(\frac{x-n}{n}\right)!}{\left(\frac{1}{n}\right)!}\\
& =x\left(x-n\right)!^{n}
\end{align*}
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多重階乗と拡張多重階乗の定義
\[
\left(x\right)!^{n}=n^{\frac{x-1}{n}}\frac{\left(\frac{x}{n}\right)!}{\left(\frac{1}{n}\right)!}
\]
ウォリス積分の拡張2重階乗表示
\[
\int_{0}^{\frac{\pi}{2}}\sin^{n}\theta d\theta=\frac{\left(n-1\right)!^{2}}{\left(n\right)!^{2}}\sqrt{\frac{\pi}{2}}
\]
(拡張)多重階乗と階乗の関係
\[
\left(an+b\right)!_{a}=\frac{a^{n}b!_{a}\left(n+\frac{b}{a}\right)!}{\left(\frac{b}{a}\right)!}
\]
階乗の多重階乗表示
\[
n!=\prod_{k=0}^{j-1}\left(n-k\right)!_{j}
\]