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*}
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2項係数の総和その他
\[
\sum_{k=1}^{n-1}\frac{C\left(k-n,k\right)}{k}=-H_{n-1}
\]
2項係数の関係その他
\[
C\left(\alpha,\beta\right)C\left(\beta,\gamma\right)=C\left(\alpha,\gamma\right)C\left(\alpha-\gamma,\beta-\gamma\right)
\]
一般ヴァンデルモンドの畳み込み定理
\[
\sum_{k_{1}+\cdots+k_{p}=m}\prod_{j=1}^{p}C\left(n_{j},k_{j}\right)=C\left(\sum_{j=1}^{p}n_{j},m\right)
\]
2項係数の第1引数と第2引数同士の総和
\[
\sum_{j=0}^{k-a}\left(-1\right)^{j}C\left(k,j+a\right)C\left(j+b,c\right)=\begin{cases}
\left(-1\right)^{k-a}C\left(b-a,c-k\right) & a-b+c\leq k\\
0 & k<a-b+c
\end{cases}
\]