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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2項係数を含む総和

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