階乗冪(下降階乗・上昇階乗)の微分
(1)
\begin{align*} \frac{d}{dx}P(x,y) & =P(x,y)\left\{ \psi(1+x)-\psi(1+x-y)\right\} \\ & =P(x,y)\left(H_{x}-H_{x-y}\right) \end{align*}(2)
\[ \frac{d}{dy}P(x,y)=P(x,y)\psi(1+x-y) \](3)
\begin{align*} \frac{d}{dx}Q(x,y) & =Q(x,y)\left\{ \psi(x+y)-\psi(x)\right\} \\ & =Q(x,y)\left(H_{x+y-1}-H_{x-1}\right) \end{align*}(4)
\[ \frac{d}{dy}Q(x,y)=Q(x,y)\psi(x+y) \](1)
\begin{align*} \frac{d}{dx}P(x,y) & =\frac{d}{dx}\frac{\Gamma(1+x)}{\Gamma(1+x-y)}\\ & =\frac{\Gamma(1+x)\psi(1+x)}{\Gamma(1+x-y)}-\frac{\Gamma(1+x)\Gamma(1+x-y)\psi(1+x-y)}{\Gamma^{2}(1+x-y)}\\ & =P(x,y)\left\{ \psi(1+x)-\psi(1+x-y)\right\} \qquad(*)\\ & =P(x,y)\left\{ \left(-\gamma+H_{x}\right)-\left(-\gamma+H_{x-y}\right)\right\} \\ & =P(x,y)\left(H_{x}-H_{x-y}\right) \end{align*}(2)
\begin{align*} \frac{d}{dy}P(x,y) & =\frac{d}{dy}\frac{\Gamma(1+x)}{\Gamma(1+x-y)}\\ & =\frac{\Gamma(1+x)\Gamma(1+x-y)\psi(1+x-y)}{\Gamma^{2}(1+x-y)}\\ & =P(x,y)\psi(1+x-y) \end{align*}(3)
\begin{align*} \frac{d}{dx}Q(x,y) & =\frac{d}{dx}\frac{\Gamma(x+y)}{\Gamma(x)}\\ & =\frac{\Gamma(x+y)\psi(x+y)}{\Gamma(x)}-\frac{\Gamma(x+y)\Gamma(x)\psi(x)}{\Gamma^{2}(x)}\\ & =Q(x,y)\left\{ \psi(x+y)-\psi(x)\right\} \qquad(*)\\ & =Q(x,y)\left\{ \left(-\gamma+H_{x+y-1}\right)-\left(-\gamma+H_{x-1}\right)\right\} \\ & =Q(x,y)\left(H_{x+y-1}-H_{x-1}\right) \end{align*}(4)
\begin{align*} \frac{d}{dy}Q(x,y) & =\frac{d}{dy}\frac{\Gamma(x+y)}{\Gamma(x)}\\ & =\frac{\Gamma(x+y)\psi(x+y)}{\Gamma(x)}\\ & =Q(x,y)\psi(x+y) \end{align*}ページ情報
| タイトル | 階乗冪(下降階乗・上昇階乗)の微分 |
| URL | https://www.nomuramath.com/aicwvyan/ |
| SNSボタン |
階乗冪(上昇階乗・下降階乗)の１項間漸化式
\[
P(x+1,y)=\frac{x+1}{x-y+1}P(x,y)
\]
階乗冪(上昇階乗・下降階乗)同士の関係
\[
P(x,y)=P^{-1}(x-y,-y)
\]
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)
\[
P(x+y,n)=\sum_{k=0}^{n}C(n,k)P(x,k)P(y,n-k)
\]