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。
\(n\in\mathbb{Z}\)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*}ページ情報
| タイトル | 階乗冪(上昇階乗・下降階乗)同士の関係 |
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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
\[
\forall m,n\in\mathbb{Z},0\leq m<n\Leftrightarrow P\left(m,n\right)=0
\]
階乗冪(下降階乗・上昇階乗)の1/2値
\[
P\left(-\frac{1}{2},n\right)=\frac{(-1)^{n}(2n-1)!}{2^{2n-1}(n-1)!}
\]
階乗冪(下降階乗・上昇階乗)の差分
\[
P(x,y)=\frac{1}{y+1}\left(P(x+1,y+1)-P(x,y+1)\right)
\]