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(下降階乗・上昇階乗)の和分
\[
\sum_{k=1}^{m}P(k,n)=\frac{1}{n+1}P(m+1,n+1)
\]
和の階乗冪(下降階乗・上昇階乗)
\[
P(x+y,n)=\sum_{k=0}^{n}C(n,k)P(x,k)P(y,n-k)
\]
階乗冪(下降階乗・上昇階乗)の差分
\[
P(x,y)=\frac{1}{y+1}\left(P(x+1,y+1)-P(x,y+1)\right)
\]
階乗冪(上昇階乗・下降階乗)とその逆数の値が0となるとき
\[
\forall m,n\in\mathbb{Z},0\leq m<n\Leftrightarrow P\left(m,n\right)=0
\]