階乗・ガンマ関数の商と階乗冪(上昇階乗・下降階乗)の関係
階乗・ガンマ関数の商と階乗冪(上昇階乗・下降階乗)の関係
(1)
\begin{align*} \frac{x!}{y!} & =P\left(x,x-y\right)\\ & =P^{-1}\left(y,y-x\right)\\ & =Q\left(y+1,x-y\right)\\ & =Q^{-1}\left(x+1,y-x\right) \end{align*}(2)
\begin{align*} \frac{\Gamma\left(x\right)}{\Gamma\left(y\right)} & =P\left(x-1,x-y\right)\\ & =P^{-1}\left(y-1,y-x\right)\\ & =Q\left(y,x-y\right)\\ & =Q^{-1}\left(x,y-x\right) \end{align*}(1)
\begin{align*} \frac{x!}{y!} & =\frac{x!}{\left(x-\left(x-y\right)\right)!}\\ & =P\left(x,x-y\right) \end{align*} \begin{align*} \frac{x!}{y!} & =\frac{\left(y-\left(y-x\right)\right)!}{y!}\\ & =P^{-1}\left(y,y-x\right) \end{align*} \begin{align*} \frac{x!}{y!} & =\frac{\Gamma\left(x+1\right)}{\Gamma\left(y+1\right)}\\ & =\frac{\Gamma\left(y+1+\left(x-y\right)\right)}{\Gamma\left(y+1\right)}\\ & =Q\left(y+1,x-y\right) \end{align*} \begin{align*} \frac{x!}{y!} & =\frac{\Gamma\left(x+1\right)}{\Gamma\left(y+1\right)}\\ & =\frac{\Gamma\left(x+1\right)}{\Gamma\left(x+1+\left(y-x\right)\right)}\\ & =Q^{-1}\left(x+1,y-x\right) \end{align*}(2)
(1)より、\begin{align*} \frac{\Gamma\left(x\right)}{\Gamma\left(y\right)} & =\frac{\left(x-1\right)!}{\left(y-1\right)!}\\ & =P\left(x-1,x-y\right)\\ & =P^{-1}\left(y-1,y-x\right)\\ & =Q\left(y,x-y\right)\\ & =Q^{-1}\left(x,y-x\right) \end{align*}
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| タイトル | 階乗・ガンマ関数の商と階乗冪(上昇階乗・下降階乗)の関係 |
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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)とその逆数の値が0となるとき
\[
\forall m,n\in\mathbb{Z},0\leq m<n\Leftrightarrow P\left(m,n\right)=0
\]
階乗冪(下降階乗・上昇階乗)の差分
\[
P(x,y)=\frac{1}{y+1}\left(P(x+1,y+1)-P(x,y+1)\right)
\]
和の階乗冪(下降階乗・上昇階乗)
\[
P(x+y,n)=\sum_{k=0}^{n}C(n,k)P(x,k)P(y,n-k)
\]