階乗冪(上昇階乗・下降階乗)の１項間漸化式
(1)
\[ P(x+1,y)=\frac{x+1}{x-y+1}P(x,y) \]
(2)
\[ P(x-1,y)=\frac{x-y}{x}P(x,y) \]
(3)
\[ P(x,y+1)=(x-y)P(x,y) \]
(4)
\[ P(x,y-1)=\frac{1}{x-y+1}P(x,y) \]
(5)
\[ Q(x+1,y)=\frac{x+y}{x}Q(x,y) \]
(6)
\[ Q(x-1,y)=\frac{x-1}{x+y-1}Q(x,y) \]
(7)
\[ Q(x,y+1)=(x+y)Q(x,y) \]
(8)
\[ Q(x,y-1)=\frac{1}{x+y-1}Q(x,y) \]
(1)
\begin{align*} P(x+1,y) & =\frac{(x+1)!}{(x+1-y)!}\\ & =\frac{x+1}{x+1-y}\frac{x!}{(x-y)!}\\ & =\frac{x+1}{x-y+1}P(x,y) \end{align*}
(2)
\begin{align*} P(x-1,y) & =\frac{(x-1)!}{(x-1-y)!}\\ & =\frac{x-y}{x}\frac{x!}{(x-y)!}\\ & =\frac{x-y}{x}P(x,y) \end{align*}
(3)
\begin{align*} P(x,y+1) & =\frac{x!}{(x-y-1)!}\\ & =(x-y)\frac{x!}{(x-y)!}\\ & =(x-y)P(x,y) \end{align*}
(4)
\begin{align*} P(x,y-1) & =\frac{x!}{(x-y+1)!}\\ & =\frac{1}{x-y+1}\frac{x!}{(x-y)!}\\ & =\frac{1}{x-y+1}P(x,y) \end{align*}
(5)
\begin{align*} Q(x+1,y) & =\frac{(x+1+y-1)!}{(x+1-1)!}\\ & =\frac{x+y}{x}\frac{(x+y-1)!}{(x-1)!}\\ & =\frac{x+y}{x}Q(x,y) \end{align*}
(6)
\begin{align*} Q(x-1,y) & =\frac{(x-1+y-1)!}{(x-1-1)!}\\ & =\frac{x-1}{x+y-1}\frac{(x+y-1)!}{(x-1)!}\\ & =\frac{x-1}{x+y-1}Q(x,y) \end{align*}
(7)
\begin{align*} Q(x,y+1) & =\frac{(x+y+1-1)!}{(x-1)!}\\ & =(x+y)\frac{(x+y-1)!}{(x-1)!}\\ & =(x+y)Q(x,y) \end{align*}
(8)
\begin{align*} Q(x,y-1) & =\frac{(x+y-1-1)!}{(x-1)!}\\ & =\frac{1}{x+y-1}\frac{(x+y-1)!}{(x-1)!}\\ & =\frac{1}{x+y-1}Q(x,y) \end{align*}
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