階乗冪(上昇階乗・下降階乗)同士の関係

(1)

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

(2)

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

(3)

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

(4)

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

(5)

\[ P(x,n)=(-1)^{n}Q(-x,n)\qquad,\qquad n\in\mathbb{Z} \]

(6)

\[ Q(x,n)=(-1)^{n}P(-x,n)\qquad,\qquad n\in\mathbb{Z} \]

(7)

\[ P(x,n)=(-1)^{n}P(-x+n-1,n)\qquad,\qquad n\in\mathbb{Z} \]

(8)

\[ Q(x,n)=(-1)^{n}Q(-x-n+1,n)\qquad,\qquad n\in\mathbb{Z} \]

(1)

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

(2)

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

(3)

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

(4)

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

(5)

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

(5)-2

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

(6)

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

(6)-2

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

(7)

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

(7)-2

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

(8)

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

(8)-2

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