階乗冪(上昇階乗・下降階乗)同士の関係

階乗冪(上昇階乗・下降階乗)同士の関係

\(n\in\mathbb{Z}\)とする。

(1)

\[ P\left(x,y\right)=P^{-1}\left(x-y,-y\right) \]

(2)

\[ Q\left(x,y\right)=Q^{-1}\left(x+y,-y\right) \]

(3)

\[ P\left(x,y\right)=Q\left(x-y+1,y\right) \]

(4)

\[ Q\left(x,y\right)=P\left(x+y-1,y\right) \]

(5)

\[ P\left(x,-y\right)=Q^{-1}\left(x+1,y\right) \]

(6)

\[ Q\left(x,-y\right)=P^{-1}\left(x-1,y\right) \]

(7)

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

(8)

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

(9)

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

(10)

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

(11)

\begin{align*} P\left(-x,-n\right) & =\left(-1\right)^{n}P^{-1}\left(x-1,n\right)\\ & =\left(-1\right)^{n}Q^{-1}\left(x-n,n\right) \end{align*}

(12)

\begin{align*} Q\left(-x,-n\right) & =\left(-1\right)^{n}Q^{-1}\left(x+1,n\right)\\ & =\left(-1\right)^{n}P^{-1}\left(x+n,n\right) \end{align*}

(1)

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

(2)

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

(3)

\begin{align*} P\left(x,y\right) & =\frac{x!}{\left(x-y\right)!}\\ & =Q\left(x-y+1,y\right) \end{align*}

(4)

\begin{align*} Q\left(x,y\right) & =\frac{\Gamma\left(x+y\right)}{\Gamma\left(x\right)}\\ & =\frac{\left(x+y-1\right)!}{\left(x+y-1-y\right)!}\\ & =P\left(x+y-1,y\right) \end{align*}

(5)

\begin{align*} P\left(x,-y\right) & =Q\left(x+y+1,-y\right)\\ & =Q^{-1}\left(x+1,y\right) \end{align*}

(6)

\begin{align*} Q\left(x,-y\right) & =P\left(x-y-1,-y\right)\\ & =P^{-1}\left(x-1,y\right) \end{align*}

(7)

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

(7)-2

\begin{align*} P\left(x,n\right) & =\left(-1\right)^{n}P\left(-x+n-1,n\right)\qquad,\qquad\text{(7)より}\\ & =\left(-1\right)^{n}Q\left(-x,n\right) \end{align*}

(8)

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

(8)-2

\begin{align*} Q\left(x,n\right) & =\left(-1\right)^{n}Q\left(-x-n+1,n\right)\qquad,\qquad\text{(8)より}\\ & =\left(-1\right)^{n}P\left(-x,n\right) \end{align*}

(9)

\begin{align*} P\left(x,n\right) & =\frac{\Gamma\left(1+x\right)}{\Gamma\left(1+x-n\right)}\\ & =\frac{\Gamma\left(n-x\right))\sin\left(\left(n-x\right)\pi\right)}{\Gamma\left(-x\right)\sin\left(-x\pi\right)}\\ & =\frac{\Gamma\left(n-x\right)\left\{ \sin\left(n\pi\right)\cos\left(-x\pi\right)+\cos\left(n\pi\right)\sin\left(-x\pi\right)\right\} }{\Gamma\left(-x\right)\sin\left(-x\pi\right)}\\ & =\left(-1\right)^{n}\frac{\Gamma\left(n-x\right)}{\Gamma\left(-x\right)}\\ & =\left(-1\right)^{n}P\left(-x+n-1,n\right) \end{align*}

(9)-2

\begin{align*} P\left(x,n\right) & =\left(-1\right)^{n}Q\left(-x,n\right)\\ & =\left(-1\right)^{n}P\left(-x+n-1,n\right) \end{align*}

(10)

\begin{align*} Q\left(x,n\right) & =\frac{\Gamma\left(1+x+n-1\right)}{\Gamma\left(1+x-1\right)}\\ & =\frac{\Gamma\left(1-x\right)\sin\left(x\pi\right)}{\Gamma\left(1-x-n\right)\sin\left(\left(x+n\right)\pi\right)}\\ & =\frac{\Gamma\left(1-x\right)\sin\left(x\pi\right)}{\Gamma\left(1-x-n\right))\left\{ \sin\left(x\pi\right)\cos\left(n\pi\right)+\cos\left(x\pi\right)\sin\left(n\pi\right)\right\} }\\ & =\left(-1\right)^{n}\frac{\Gamma\left(1-x\right)}{\Gamma\left(1-x-n\right)}\\ & =\left(-1\right)^{n}Q\left(-x-n+1,n\right) \end{align*}

(10)-2

\begin{align*} Q\left(x,n\right) & =\left(-1\right)^{n}P\left(-x,n\right)\\ & =\left(-1\right)^{n}Q\left(-x-n+1,n\right) \end{align*}

(11)

\begin{align*} P\left(-x,-n\right) & =Q^{-1}\left(-x+1,n\right)\\ & =\left(-1\right)^{n}P^{-1}\left(x-1,n\right)\\ & =\left(-1\right)^{n}Q^{-1}\left(x-n,n\right) \end{align*}

(12)

\begin{align*} Q\left(-x,-n\right) & =P^{-1}\left(-x-1,n\right)\\ & =\left(-1\right)^{n}Q^{-1}\left(x+1,n\right)\\ & =\left(-1\right)^{n}P^{-1}\left(x+n,n\right) \end{align*}

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階乗冪(上昇階乗・下降階乗)同士の関係

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