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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パスカルの法則の一般形
\[
C\left(x+n,y+n\right)=\sum_{k=0}^{n}C\left(n,k\right)C\left(x,y+k\right)
\]
飛び飛びの2項定理
\[
\sum_{k=0}^{\infty}C\left(n,2k\right)a^{2k}b^{n-2k}=\frac{1}{2}\left\{ \left(a+b\right)^{n}+\left(-a+b\right)^{n}\right\}
\]
2項係数の第1引数と第2引数同士の総和
\[
\sum_{j=0}^{k-a}\left(-1\right)^{j}C\left(k,j+a\right)C\left(j+b,c\right)=\begin{cases}
\left(-1\right)^{k-a}C\left(b-a,c-k\right) & a-b+c\leq k\\
0 & k<a-b+c
\end{cases}
\]
2項係数の逆数の差分
\[
C^{-1}(k+j+1,j+1)=\frac{j+1}{j}\left(C^{-1}(k+j,j)-C^{-1}(k+j+1,j)\right)
\]