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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| タイトル | 階乗冪(上昇階乗・下降階乗)の１項間漸化式 |
| URL | https://www.nomuramath.com/n0lg5oq0/ |
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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
\[
\forall m,n\in\mathbb{Z},0\leq m<n\Leftrightarrow P\left(m,n\right)=0
\]
和の階乗冪(下降階乗・上昇階乗)
\[
P(x+y,n)=\sum_{k=0}^{n}C(n,k)P(x,k)P(y,n-k)
\]
階乗・ガンマ関数の商と階乗冪(上昇階乗・下降階乗)の関係
\[
\frac{\Gamma\left(x\right)}{\Gamma\left(y\right)}=Q\left(y,x-y\right)
\]
階乗冪(上昇階乗・下降階乗)の母関数
\[
\sum_{k=0}^{\infty}P(k,n)x^{k}=\frac{x^{n}n!}{(1-x)^{n+1}}
\]