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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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)多重階乗の逆数和
\[
\sum_{k=0}^{n}\frac{1}{\left(ak+b\right)!_{a}}=\frac{e^{\frac{1}{a}}a^{\frac{b}{a}}\Gamma\left(\frac{b}{a}+1\right)}{b!_{a}}\left(\frac{\Gamma\left(n+\frac{b}{a}+1,\frac{1}{a}\right)}{\Gamma\left(n+\frac{b}{a}+1\right)}-\frac{\Gamma\left(\frac{b}{a},\frac{1}{a}\right)}{\Gamma\left(\frac{b}{a}\right)}\right)
\]
(拡張)多重階乗と階乗の関係
\[
\left(an+b\right)!_{a}=\frac{a^{n}b!_{a}\left(n+\frac{b}{a}\right)!}{\left(\frac{b}{a}\right)!}
\]
多重階乗と拡張多重階乗の定義
\[
\left(x\right)!^{n}=n^{\frac{x-1}{n}}\frac{\left(\frac{x}{n}\right)!}{\left(\frac{1}{n}\right)!}
\]