(拡張)多重階乗の逆数和
(拡張)多重階乗の逆数和
\(n\in\mathbb{N}\)とする。
\(n\in\mathbb{N}\)とする。
(1)
\[ \sum_{k=0}^{n}\frac{1}{\left(ak+b\right)!_{a}}=\frac{e^{\frac{1}{a}}a^{\frac{b}{a}}\Gamma\left(\frac{b}{a}+1\right)}{b!_{a}}\left(\frac{\Gamma\left(n+\frac{b}{a}+1,\frac{1}{a}\right)}{\Gamma\left(n+\frac{b}{a}+1\right)}-\frac{\Gamma\left(\frac{b}{a},\frac{1}{a}\right)}{\Gamma\left(\frac{b}{a}\right)}\right) \](2)
\[ \sum_{k=0}^{\infty}\frac{1}{\left(ak+b\right)!_{a}}=\frac{e^{\frac{1}{a}}a^{\frac{b}{a}}\Gamma\left(\frac{b}{a}+1\right)}{b!_{a}}\left(1-\frac{\Gamma\left(\frac{b}{a},\frac{1}{a}\right)}{\Gamma\left(\frac{b}{a}\right)}\right) \](3)
\[ \sum_{k=0}^{n}\frac{1}{\left(ak+b\right)!^{a}}=\frac{e^{\frac{1}{a}}a^{\frac{b}{a}}\Gamma\left(\frac{b}{a}+1\right)}{b!^{a}}\left(\frac{\Gamma\left(n+\frac{b}{a}+1,\frac{1}{a}\right)}{\Gamma\left(n+\frac{b}{a}+1\right)}-\frac{\Gamma\left(\frac{b}{a},\frac{1}{a}\right)}{\Gamma\left(\frac{b}{a}\right)}\right) \](4)
\[ \sum_{k=0}^{\infty}\frac{1}{\left(ak+b\right)!^{a}}=\frac{e^{\frac{1}{a}}a^{\frac{b}{a}}\Gamma\left(\frac{b}{a}+1\right)}{b!^{a}}\left(1-\frac{\Gamma\left(\frac{b}{a},\frac{1}{a}\right)}{\Gamma\left(\frac{b}{a}\right)}\right) \]-
\(\Gamma\left(x\right)\)はガンマ関数、\(\Gamma\left(k,x\right)\)は第2種不完全ガンマ関数、\(n!_{p}\)は多重階乗、\(n!^{p}\)は拡張多重階乗(1)
\begin{align*} \sum_{k=0}^{n}\frac{1}{\left(ak+b\right)!_{a}} & =\sum_{k=0}^{n}\frac{1}{m^{k}b!_{a}}\frac{\Gamma\left(\frac{b}{a}+1\right)}{\left(k+\frac{b}{a}\right)!}\\ & =\frac{\Gamma\left(\frac{b}{a}+1\right)}{b!_{a}}\sum_{k=0}^{n}\frac{1}{a^{k}\left(k+\frac{b}{a}\right)!}\\ & =\frac{\Gamma\left(\frac{b}{a}+1\right)a^{\frac{b}{a}}}{b!_{a}}\sum_{k=0}^{n}\frac{1}{\left(k+\frac{b}{a}\right)!}\left(\frac{1}{a}\right)^{k+\frac{b}{a}}\\ & =\frac{\Gamma\left(\frac{b}{a}+1\right)a^{\frac{b}{a}}}{b!_{a}}\sum_{k=0}^{n}e^{\frac{1}{a}}\left(\frac{\Gamma\left(k+\frac{b}{a}+1,\frac{1}{a}\right)}{\Gamma\left(k+\frac{b}{a}+1\right)}-\frac{\Gamma\left(k+\frac{b}{a},\frac{1}{a}\right)}{\Gamma\left(k+\frac{b}{a}\right)}\right)\\ & =\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) \end{align*}(2)
\begin{align*} \sum_{k=0}^{\infty}\frac{1}{\left(ak+b\right)!_{a}} & =\lim_{n\rightarrow\infty}\sum_{k=0}^{n}\frac{1}{\left(ak+b\right)!_{a}}\\ & =\lim_{n\rightarrow\infty}\frac{e^{\frac{1}{a}}\Gamma\left(\frac{b}{a}+1\right)a^{\frac{b}{a}}}{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)\\ & =\frac{e^{\frac{1}{a}}a^{\frac{b}{a}}\Gamma\left(\frac{b}{a}+1\right)}{b!_{a}}\left(1-\frac{\Gamma\left(\frac{b}{a},\frac{1}{a}\right)}{\Gamma\left(\frac{b}{a}\right)}\right) \end{align*}(3)
(1)と同じ(4)
(2)と同じ
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(拡張)多重階乗と階乗の関係
\[
\left(an+b\right)!_{a}=\frac{a^{n}b!_{a}\left(n+\frac{b}{a}\right)!}{\left(\frac{b}{a}\right)!}
\]
拡張多重階乗の簡単な値
\[
0!^{n}=\frac{1}{\sqrt[n]{n}\left(\frac{1}{n}\right)!}
\]
多重階乗と拡張多重階乗の定義
\[
\left(x\right)!^{n}=n^{\frac{x-1}{n}}\frac{\left(\frac{x}{n}\right)!}{\left(\frac{1}{n}\right)!}
\]
多重階乗の階乗表示
\[
\left(qn+r\right)!_{n}=r!_{n}n^{q}\frac{\left(q+\frac{r}{n}\right)!}{\left(\frac{r}{n}\right)!}
\]