拡張多重階乗の漸化式
拡張多重階乗の漸化式
\[ x!^{n}=x\left(x-n\right)!^{n} \]
*
\(x!^{n}\)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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(拡張)多重階乗と階乗の関係
\[
\left(an+b\right)!_{a}=\frac{a^{n}b!_{a}\left(n+\frac{b}{a}\right)!}{\left(\frac{b}{a}\right)!}
\]
多重階乗同士の関係
\[
\left(qn+r\right)!^{n}=r!^{n}\frac{\left(qn+r\right)!_{n}}{r!_{n}}
\]
負の多重階乗
\[
\left(-\left(qn+r\right)\right)!_{n}=\frac{\left(-1\right)^{q}}{\left(qn-\left(n-r\right)\right)!_{n}}
\]
拡張多重階乗の簡単な値
\[
0!^{n}=\frac{1}{\sqrt[n]{n}\left(\frac{1}{n}\right)!}
\]