1次式の総乗と階乗
1次式の総乗と階乗
\(a,b\in\mathbb{Z}\)とする。
\[ \prod_{k=a}^{b}\left(kn+r\right)=n^{b-a+1}\frac{\left(b+\frac{r}{n}\right)!}{\Gamma\left(a+\frac{r}{n}\right)} \]
\(a,b\in\mathbb{Z}\)とする。
\[ \prod_{k=a}^{b}\left(kn+r\right)=n^{b-a+1}\frac{\left(b+\frac{r}{n}\right)!}{\Gamma\left(a+\frac{r}{n}\right)} \]
(0)
\begin{align*} \prod_{k=a}^{b}\left(kn+r\right) & =n^{b-a+1}\prod_{k=a}^{b}\left(k+\frac{r}{n}\right)\\ & =n^{b-a+1}\prod_{k=a}^{b}\frac{\left(k+\frac{r}{n}\right)!}{\left(k+\frac{r}{n}-1\right)!}\\ & =n^{b-a+1}\frac{\left(b+\frac{r}{n}\right)!}{\Gamma\left(a+\frac{r}{n}\right)} \end{align*}(0)-2
\begin{align*} \prod_{k=a}^{b}\left(kn+r\right) & =\prod_{k=a}^{-1}\left(kn+r\right)\prod_{k=0}^{b}\left(kn+r\right)\\ & =\prod_{k=0}^{a-1}\left(kn+r\right)^{-1}\prod_{k=0}^{b}\left(kn+r\right)\\ & =\left\{ n^{a-1}r\frac{\left(a-1+\frac{r}{n}\right)!}{\frac{r}{n}!}\right\} ^{-1}n^{b}r\frac{\left(b+\frac{r}{n}\right)!}{\frac{r}{n}!}\\ & =n^{b-a+1}\frac{\left(b+\frac{r}{n}\right)!}{\left(a+\frac{r}{n}-1\right)!}\\ & =n^{b-a+1}\frac{\left(b+\frac{r}{n}\right)!}{\Gamma\left(a+\frac{r}{n}\right)} \end{align*}
ページ情報
| タイトル | 1次式の総乗と階乗 |
| URL | https://www.nomuramath.com/f039zr6h/ |
| SNSボタン |
第2種不完全ガンマ関数とガンマ関数の比の極限
\[
\lim_{k\rightarrow0}\frac{\Gamma\left(k,x\right)}{\Gamma\left(k\right)}=\delta_{0x}
\]
階乗と階乗の逆数の母関数
\[
\frac{x^{a}}{a!}=e^{x}\left(\frac{\Gamma\left(a+1,x\right)}{\Gamma\left(a+1\right)}-\frac{\Gamma\left(a,x\right)}{\Gamma\left(a\right)}\right)
\]
ポリガンマ関数同士の差の極限
\[
\lim_{z\rightarrow0}\left(\psi^{\left(n\right)}\left(z-m\right)-\psi^{\left(n\right)}\left(z\right)\right)=n!H_{m,n+1}
\]
ガンマ関数のハンケル積分表示
\[
\Gamma\left(z\right)=\frac{i}{2\sin\left(\pi z\right)}\int_{C}\left(-\tau\right)^{z-1}e^{-\tau}d\tau
\]