ベータ関数と2項係数の逆数の級数表示
(1)ベータ関数の級数表示
\[ B(x,y)=\sum_{k=0}^{\infty}\frac{C(k-y,k)}{x+k} \](2)2項係数の逆数の級数表示
\[ \frac{1}{C(x,y)}=\sum_{k=0}^{\infty}(-1)^{k}C(y,k)+\sum_{k=1}^{\infty}(-1)^{k-1}C(y,k)\frac{k}{x-y+k} \](3)
\(n\in\mathbb{N}\)とする。\[ \frac{1}{C(x,n)}=\sum_{k=0}^{n-1}(-1)^{n-k-1}C(n,k)\frac{n-k}{x-k} \]
(4)
\(n\in\mathbb{N}\)とする。\[ \frac{1}{C(x+n,n)}=\sum_{k=1}^{n}(-1)^{k-1}C(n,k)\frac{k}{x+k} \]
(1)
\begin{align*} B(x,y) & =\int_{0}^{1}t^{x-1}(1-t)^{y-1}dt\\ & =\sum_{k=0}^{\infty}C(y-1,k)\int_{0}^{1}t^{x-1}(-t)^{k}dt\\ & =\sum_{k=0}^{\infty}C(y-1,k)(-1)^{k}\int_{0}^{1}t^{x+k-1}dt\\ & =\sum_{k=0}^{\infty}C(y-1,k)(-1)^{k}\left[\frac{t^{x+k}}{x+k}\right]_{t=0}^{t=1}\\ & =\sum_{k=0}^{\infty}(-1)^{k}\frac{C(y-1,k)}{x+k}\\ & =\sum_{k=0}^{\infty}(-1)^{k}\frac{P(y-1,k)}{k!\left(x+k\right)}\\ & =\sum_{k=0}^{\infty}\frac{Q(1-y,k)}{k!\left(x+k\right)}\\ & =\sum_{k=0}^{\infty}\frac{P(k-y,k)}{k!\left(x+k\right)}\\ & =\sum_{k=0}^{\infty}\frac{C(k-y,k)}{x+k} \end{align*}(2)
\begin{align*} \frac{1}{C(x,y)} & =(x+1)B(x-y+1,y+1)\\ & =(x+1)\sum_{k=0}^{\infty}(-1)^{k}\frac{C(y,k)}{x-y+1+k}\\ & =\sum_{k=0}^{\infty}(-1)^{k}C(y,k)\left(1+\frac{y-k}{x-y+1+k}\right)\\ & =\sum_{k=0}^{\infty}(-1)^{k}C(y,k)+\sum_{k=0}^{\infty}(-1)^{k}C(y,k)\frac{y-k}{x-y+1+k}\\ & =\sum_{k=0}^{\infty}(-1)^{k}C(y,k)+\sum_{k=0}^{\infty}(-1)^{k}C(y,k+1)\frac{k+1}{y-k}\frac{y-k}{x-y+1+k}\\ & =\sum_{k=0}^{\infty}(-1)^{k}C(y,k)+\sum_{k=1}^{\infty}(-1)^{k-1}C(y,k)\frac{k}{x-y+k} \end{align*}(3)
\begin{align*} \frac{1}{C(x,n)} & =\sum_{k=0}^{\infty}(-1)^{k}C(n,k)+\sum_{k=1}^{\infty}(-1)^{k-1}C(n,k)\frac{k}{x-n+k}\\ & =\sum_{k=1}^{\infty}(-1)^{k-1}C(n,k)\frac{k}{x-n+k}\\ & =\sum_{j=0}^{n-1}(-1)^{n-j-1}C(n,j)\frac{n-j}{x-j}\cmt{k=n-j} \end{align*}(4)
\begin{align*} \frac{1}{C(x+n,n)} & =\sum_{k=0}^{n-1}(-1)^{n-k-1}C(n,k)\frac{n-k}{x+n-k}\\ & =\sum_{k=1}^{n}(-1)^{k-1}C(n,k)\frac{k}{x+k}\cmt{k\rightarrow n-k} \end{align*}ページ情報
タイトル | ベータ関数と2項係数の逆数の級数表示 |
URL | https://www.nomuramath.com/b94maeix/ |
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ベータ関数になる積分
\[
\int_{0}^{\frac{\pi}{2}}\sin^{x}t\cos^{y}tdt=\frac{1}{2}B\left(\frac{x+1}{2},\frac{y+1}{2}\right)
\]
ベータ関数の微分
\[
\frac{\partial}{\partial x}B(x,y)=B(x,y)\left\{ \psi(x)-\psi(x+y)\right\}
\]
ベータ関数の関数等式
\[
xB(x,y+1)=yB(x+1,y)
\]
ベータ関数とガンマ関数の関係
\[
B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
\]