多重階乗同士の関係

多重階乗同士の関係

\(q\in\mathbb{N}_{0}\)とする。

(1)

\[ \left(qn+1\right)!_{n}=\left(qn+1\right)!^{n} \]

(2)

\[ \left(qn+r\right)!^{n}=r!^{n}\frac{\left(qn+r\right)!_{n}}{r!_{n}} \]

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\(x!_{n}\)は多重階乗、\(x!^{n}\)は拡張多重階乗。

(1)

\begin{align*} \left(qn+1\right)!_{n} & =n^{q}\frac{\left(q+\frac{1}{n}\right)!}{\left(\frac{1}{n}\right)!}\\ & =n^{\frac{qn+1-1}{n}}\frac{\left(\frac{qn+1}{n}\right)!}{\left(\frac{1}{n}\right)!}\\ & =\left(qn+1\right)!^{n} \end{align*}

(2)

\begin{align*} \left(qn+r\right)!^{n} & =r!^{n}\prod_{k=1}^{q}\frac{\left(kn+r\right)!^{n}}{\left(kn+r-n\right)!^{n}}\\ & =r!^{n}\prod_{k=1}^{q}\left(kn+r\right)\\ & =r!^{n}\prod_{k=1}^{q}\frac{\left(kn+r\right)!_{n}}{\left(kn+r-n\right)!_{n}}\\ & =r!^{n}\frac{\left(qn+r\right)!_{n}}{r!_{n}} \end{align*}

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多重階乗同士の関係

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