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)同士の関係
\[
P(x,y)=P^{-1}(x-y,-y)
\]
階乗冪(下降階乗・上昇階乗)の1/2値
\[
P\left(-\frac{1}{2},n\right)=\frac{(-1)^{n}(2n-1)!}{2^{2n-1}(n-1)!}
\]
階乗冪(上昇階乗・下降階乗)の１項間漸化式
\[
P(x+1,y)=\frac{x+1}{x-y+1}P(x,y)
\]