2項係数の微分
2項係数の微分
(1)
\begin{align*} \frac{d}{dx}C(x,y) & =C(x,y)\left(\psi(1+x)-\psi(1+x-y)\right)\\ & =C(x,y)\left(H_{x}-H_{x-y}\right) \end{align*}(2)
\begin{align*} \frac{d}{dy}C(x,y) & =C(x,y)\left\{ \psi(1+x-y)-\psi(1+y)\right\} \\ & =C(x,y)\left\{ H_{x-y}-H_{y}\right\} \end{align*}(1)
\begin{align*} \frac{d}{dx}C(x,y) & =\frac{1}{y!}\frac{d}{dx}P(x,y)\\ & =\frac{1}{y!}P(x,y)\left(\psi(1+x)-\psi(1+x-y)\right)\\ & =C(x,y)\left(\psi(1+x)-\psi(1+x-y)\right)\\ & =C(x,y)\left(H_{x}-H_{x-y}\right) \end{align*}(2)
\begin{align*} \frac{d}{dy}C(x,y) & =\frac{1}{y!}\frac{d}{dy}P(x,y)+P(x,y)\frac{d}{dy}\frac{1}{\Gamma(y+1)}\\ & =\frac{1}{y!}P(x,y)\psi(1+x-y)+P(x,y)\frac{-\Gamma(y+1)\psi(y+1)}{\Gamma^{2}(y+1)}\\ & =C(x,y)\psi(1+x-y)-C(x,y)\psi(y+1)\\ & =C(x,y)\left\{ \psi(1+x-y)-\psi(1+y)\right\} \\ & =C(x,y)\left\{ H_{x-y}-H_{y}\right\} \end{align*}ページ情報
| タイトル | 2項係数の微分 |
| URL | https://www.nomuramath.com/xqn5ejgc/ |
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中央2項係数の値
\[
C\left(2n,n\right)=4^{n}\left(-1\right)^{n}C\left(-\frac{1}{2},n\right)
\]
負の整数の2項係数
\[
C\left(-m,-n\right)=\left(-1\right)^{m-n}C\left(n-1,m-1\right)
\]
2項係数の2乗和
\[
\sum_{j=0}^{m}C^{2}(m,j)=C(2m,m)
\]
パスカルの法則の一般形
\[
C\left(x+n,y+n\right)=\sum_{k=0}^{n}C\left(n,k\right)C\left(x,y+k\right)
\]