階乗冪(上昇階乗・下降階乗)の定義

階乗冪(上昇階乗・下降階乗)の定義

(1)下降階乗

\[ P\left(x,y\right)=\frac{x!}{\left(x-y\right)!} \]

(2)上昇階乗

\[ Q\left(x,y\right)=\frac{\Gamma\left(x+y\right)!}{\Gamma\left(x\right)!} \]

階乗冪(上昇階乗・下降階乗)と総乗
\(n\in\mathbb{Z}\)とする。

(1)

\[ P\left(x,n\right)=\prod_{k=0}^{n-1}\left(x-k\right) \]

(2)

\[ Q\left(x,n\right)=\prod_{k=0}^{n-1}\left(x+k\right) \]

(1)

\begin{align*} P\left(x,n\right) & =\frac{x!}{\left(x-n\right)!}\\ & =\prod_{k=0}^{n-1}\left(x-k\right) \end{align*}

(2)

\begin{align*} Q\left(x,n\right) & =\frac{\Gamma\left(x+y\right)!}{\Gamma\left(x\right)!}\\ & =\frac{\left(x+n-1\right)!}{\left(x-1\right)!}\\ & =\prod_{k=0}^{n-1}(x+n-1-k)\\ & =\prod_{k=0}^{n-1}(x+k) \end{align*}

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階乗冪(上昇階乗・下降階乗)の定義

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