2項係数を含む総和
2項係数を含む総和
(1)
\[ \sum_{k=0}^{n}\frac{\left(-1\right)^{k}C\left(n,k\right)}{m+k}=\frac{1}{mC\left(m+n,m\right)} \](2)
\[ \sum_{k=0}^{n}\frac{\left(-1\right)^{k}C\left(n,k\right)}{m-k}=\frac{\left(-1\right)^{n}}{\left(m-n\right)C\left(m,n\right)} \](1)
\begin{align*} \sum_{k=0}^{n}\frac{\left(-1\right)^{k}C\left(n,k\right)}{m+k} & =\int_{0}^{1}\sum_{k=0}^{n}\left(-1\right)^{k}x^{m+k-1}C\left(n,k\right)dx\\ & =\left(-1\right)^{-n}\int_{0}^{1}x^{m-1}\sum_{k=0}^{n}\left(-1\right)^{n-k}x^{k}C\left(n,k\right)dx\\ & =\left(-1\right)^{-n}\int_{0}^{1}x^{m-1}\left(x-1\right)^{n}dx\\ & =\int_{0}^{1}x^{m-1}\left(1-x\right)^{n}dx\\ & =B\left(m,n+1\right)\\ & =\frac{\Gamma\left(m\right)\Gamma\left(n+1\right)}{\Gamma\left(m+n+1\right)}\\ & =\frac{\Gamma\left(m+1\right)\Gamma\left(n+1\right)}{m\Gamma\left(m+n+1\right)}\\ & =\frac{1}{mC\left(m+n,m\right)} \end{align*}(2)
\begin{align*} \sum_{k=0}^{n}\frac{\left(-1\right)^{k}C\left(n,k\right)}{m-k} & =\int_{0}^{1}\sum_{k=0}^{n}\left(-1\right)^{k}x^{m-k-1}C\left(n,k\right)dx\\ & =\int_{0}^{1}x^{m-n-1}\sum_{k=0}^{n}\left(-1\right)^{k}x^{n-k}C\left(n,k\right)dx\\ & =\int_{0}^{1}x^{m-n-1}\left(x-1\right)^{n}dx\\ & =\left(-1\right)^{n}\int_{0}^{1}x^{m-n-1}\left(1-x\right)^{n}dx\\ & =\left(-1\right)^{n}B\left(m-n,n+1\right)\\ & =\left(-1\right)^{n}\frac{\Gamma\left(m-n\right)\Gamma\left(n+1\right)}{\Gamma\left(m+1\right)}\\ & =\left(-1\right)^{n}\frac{\Gamma\left(m-n+1\right)\Gamma\left(n+1\right)}{\left(m-n\right)\Gamma\left(m+1\right)}\\ & =\frac{\left(-1\right)^{n}}{\left(m-n\right)C\left(m,n\right)} \end{align*}
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パスカルの法則の一般形
\[
C\left(x+n,y+n\right)=\sum_{k=0}^{n}C\left(n,k\right)C\left(x,y+k\right)
\]
2項係数の総和
\[
\sum_{k=0}^{n}P(k,m)C(n,k)=P(n,m)2^{n-m}
\]
2項係数の微分
\[
\frac{d}{dx}C(x,y) =C(x,y)\left(\psi(1+x)-\psi(1+x-y)\right)
\]
負の整数の2項係数
\[
C\left(-m,-n\right)=\left(-1\right)^{m-n}C\left(n-1,m-1\right)
\]