階乗冪(下降階乗・上昇階乗)の差分
(1)
\[ P(x,y)=\frac{1}{y+1}\left(P(x+1,y+1)-P(x,y+1)\right) \](2)
\[ \frac{1}{P(x,y)}=\frac{1}{y-1}\left(\frac{1}{P(x-1,y-1)}-\frac{1}{P(x,y-1)}\right) \](3)
\[ Q(x,y)=\frac{1}{y+1}\left(Q(x,y+1)-Q(x-1,y+1)\right) \](4)
\[ \frac{1}{Q(x,y)}=\frac{1}{y-1}\left(\frac{1}{Q(x,y-1)}-\frac{1}{Q(x+1,y-1)}\right) \](1)
\begin{align*} P(x,y) & =\frac{x!}{(x-y)!}\\ & =\frac{x!}{(x-y)!}\frac{(x+1)-(x-y)}{1+y}\\ & =\frac{1}{y+1}\left(\frac{(x+1)!}{(x-y)!}-\frac{x!}{(x-y-1)!}\right)\\ & =\frac{1}{y+1}\left(P(x+1,y+1)-P(x,y+1)\right) \end{align*}(1)-2
\begin{align*} P(x,y) & =\frac{1}{(x+1)(x-y)}P(x+1,y+2)\\ & =\frac{1}{y+1}\left(\frac{1}{x-y}-\frac{1}{x+1}\right)P(x+1,y+2)\\ & =\frac{1}{y+1}\left(P(x+1,y+1)-P(x,y+1)\right) \end{align*}(2)
\begin{align*} \frac{1}{P(x,y)} & =\frac{(x-y)!}{x!}\\ & =\frac{(x-y)!}{x!}\frac{(x)-(x-y+1)}{y-1}\\ & =\frac{1}{y-1}\left(\frac{(x-y)!}{(x-1)!}-\frac{(x-y+1)!}{x!}\right)\\ & =\frac{1}{y-1}\left(\frac{1}{P(x-1,y-1)}-\frac{1}{P(x,y-1)}\right) \end{align*}(2)-2
\begin{align*} \frac{1}{P(x,y)} & =\frac{1}{x(x-y+1)}\frac{1}{P(x-1,y-2)}\\ & =\frac{1}{y-1}\left(\frac{1}{x-y+1}-\frac{1}{x}\right)\frac{1}{P(x-1,y-2)}\\ & =\frac{1}{y-1}\left(\frac{1}{P(x-1,y-1)}-\frac{1}{P(x,y-1)}\right) \end{align*}(3)
\begin{align*} Q(x,y) & =\frac{(x+y-1)!}{(x-1)!}\\ & =\frac{(x+y-1)!}{(x-1)!}\frac{(x+y)-(x-1)}{y+1}\\ & =\frac{1}{y+1}\left(\frac{(x+y)!}{(x-1)!}-\frac{(x+y-1)!}{(x-2)!}\right)\\ & =\frac{1}{y+1}\left(Q(x,y+1)-Q(x-1,y+1)\right) \end{align*}(3)-2
\begin{align*} Q(x,y) & =\frac{1}{(x-1)(x+y)}Q(x-1,y+2)\\ & =\frac{1}{y+1}\left(\frac{1}{x-1}-\frac{1}{x+y}\right)Q(x-1,y+2)\\ & =\frac{1}{y+1}\left(Q(x,y+1)-Q(x-1,y+1)\right) \end{align*}(4)
\begin{align*} \frac{1}{Q(x,y)} & =\frac{(x-1)!}{(x+y-1)!}\\ & =\frac{(x-1)!}{(x+y-1)!}\frac{(x+y-1)-(x)}{y-1}\\ & =\frac{1}{y-1}\left(\frac{(x-1)!}{(x+y-2)!}-\frac{x!}{(x+y-1)!}\right)\\ & =\frac{1}{y-1}\left(\frac{1}{Q(x,y-1)}-\frac{1}{Q(x+1,y-1)}\right) \end{align*}(4)-2
\begin{align*} \frac{1}{Q(x,y)} & =\frac{1}{x(x+y-1)}\frac{1}{Q(x+1,y-2)}\\ & =\frac{1}{y-1}\left(\frac{1}{x}-\frac{1}{x+y-1}\right)\frac{1}{Q(x+1,y-2)}\\ & =\frac{1}{y-1}\left(\frac{1}{Q(x,y-1)}-\frac{1}{Q(x+1,y-1)}\right) \end{align*}ページ情報
| タイトル | 階乗冪(下降階乗・上昇階乗)の差分 |
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階乗冪(上昇階乗・下降階乗)の母関数
\[
\sum_{k=0}^{\infty}P(k,n)x^{k}=\frac{x^{n}n!}{(1-x)^{n+1}}
\]
階乗冪(下降階乗・上昇階乗)の指数法則
\[
P(x,y+z)=P(x,y)P(x-y,z)
\]
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１項間漸化式
\[
P(x+1,y)=\frac{x+1}{x-y+1}P(x,y)
\]