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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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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