(拡張)多重階乗と階乗の関係
(拡張)多重階乗と階乗の関係
\(n\in\mathbb{N}\;\land\;\frac{b}{a}\notin\mathbb{Z}\setminus\mathbb{N}_{0}\)とする。
\(n\in\mathbb{N}\;\land\;\frac{b}{a}\notin\mathbb{Z}\setminus\mathbb{N}_{0}\)とする。
(1)
\[ \left(an+b\right)!_{a}=\frac{a^{n}b!_{a}\left(n+\frac{b}{a}\right)!}{\left(\frac{b}{a}\right)!} \](2)
\[ \left(an+b\right)!^{a}=\frac{a^{n}b!^{a}\left(n+\frac{b}{a}\right)!}{\left(\frac{b}{a}\right)!} \]-
\(x!_{y}\)は多重階乗、\(x!^{y}\)は拡張多重階乗(1)
\begin{align*} \left(an+b\right)!_{a} & =b!_{a}\prod_{j=1}^{n}\left(aj+b\right)\\ & =a^{n}b!_{a}\prod_{j=1}^{n}\left(j+\frac{b}{a}\right)\\ & =\frac{a^{n}b!_{a}\left(n+\frac{b}{a}\right)!}{\left(\frac{b}{a}\right)!} \end{align*}(2)
(1)と同じページ情報
| タイトル | (拡張)多重階乗と階乗の関係 |
| URL | https://www.nomuramath.com/difst1p3/ |
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階乗の多重階乗表示
\[
n!=\prod_{k=0}^{j-1}\left(n-k\right)!_{j}
\]
ウォリス積分の拡張2重階乗表示
\[
\int_{0}^{\frac{\pi}{2}}\sin^{n}\theta d\theta=\frac{\left(n-1\right)!^{2}}{\left(n\right)!^{2}}\sqrt{\frac{\pi}{2}}
\]
負の多重階乗
\[
\left(-\left(qn+r\right)\right)!_{n}=\frac{\left(-1\right)^{q}}{\left(qn-\left(n-r\right)\right)!_{n}}
\]
(拡張)多重階乗の逆数和
\[
\sum_{k=0}^{n}\frac{1}{\left(ak+b\right)!_{a}}=\frac{e^{\frac{1}{a}}a^{\frac{b}{a}}\Gamma\left(\frac{b}{a}+1\right)}{b!_{a}}\left(\frac{\Gamma\left(n+\frac{b}{a}+1,\frac{1}{a}\right)}{\Gamma\left(n+\frac{b}{a}+1\right)}-\frac{\Gamma\left(\frac{b}{a},\frac{1}{a}\right)}{\Gamma\left(\frac{b}{a}\right)}\right)
\]