(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。

(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)

\[ \sum_{k=0}^{\infty}\frac{1}{\left(ak+b\right)!^{a}}=\frac{e^{\frac{1}{a}}a^{\frac{b}{a}}\Gamma\left(\frac{b}{a}+1\right)}{b!^{a}}\left(1-\frac{\Gamma\left(\frac{b}{a},\frac{1}{a}\right)}{\Gamma\left(\frac{b}{a}\right)}\right) \]

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\(\Gamma\left(x\right)\)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)多重階乗の逆数和

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