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