ポリガンマ関数同士の差の極限
ポリガンマ関数同士の差の極限
\[ \lim_{z\rightarrow0}\left(\psi^{\left(n\right)}\left(z-m\right)-\psi^{\left(n\right)}\left(z\right)\right)=n!H_{m,n+1} \]
\[ \lim_{z\rightarrow0}\left(\frac{\psi^{\left(n\right)}\left(z+\alpha\right)-\psi^{\left(n\right)}\left(z\right)}{\Gamma^{n+1}\left(z\right)}\right)=\left(-1\right)^{n}n! \]
\(H_{n,m}\)は一般化調和数
(1)
\(n,m\in\mathbb{N}_{0}\)とする。\[ \lim_{z\rightarrow0}\left(\psi^{\left(n\right)}\left(z-m\right)-\psi^{\left(n\right)}\left(z\right)\right)=n!H_{m,n+1} \]
(2)
\(n\in\mathbb{N}_{0}\)とする。\[ \lim_{z\rightarrow0}\left(\frac{\psi^{\left(n\right)}\left(z+\alpha\right)-\psi^{\left(n\right)}\left(z\right)}{\Gamma^{n+1}\left(z\right)}\right)=\left(-1\right)^{n}n! \]
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\(\psi\left(x\right)\)はポリガンマ関数\(H_{n,m}\)は一般化調和数
(1)
\begin{align*} \lim_{z\rightarrow0}\left(\psi^{\left(n\right)}\left(z-m\right)-\psi^{\left(n\right)}\left(z\right)\right) & =\lim_{z\rightarrow0}\left(\psi^{\left(n\right)}\left(z-m\right)-\psi^{\left(n\right)}\left(z+1\right)+\frac{\left(-1\right)^{n}n!}{z^{n+1}}\right)\\ & =\lim_{z\rightarrow0}\left(\psi^{\left(n\right)}\left(z+1\right)+\left(-1\right)^{n}n!\sum_{k=1}^{-m-1}\frac{1}{\left(z+k\right)^{n+1}}-\psi^{\left(n\right)}\left(z+1\right)+\frac{\left(-1\right)^{n}n!}{z^{n+1}}\right)\\ & =\lim_{z\rightarrow0}\left(-\left(-1\right)^{n}n!\sum_{k=-m}^{0}\frac{1}{\left(z+k\right)^{n+1}}+\frac{\left(-1\right)^{n}n!}{z^{n+1}}\right)\\ & =-\left(-1\right)^{n}n!\sum_{k=-m}^{-1}\frac{1}{k^{n+1}}\\ & =-\left(-1\right)^{n}n!\sum_{k=1}^{m}\frac{1}{\left(-k\right)^{n+1}}\\ & =n!\sum_{k=1}^{m}\frac{1}{k^{n+1}}\\ & =n!H_{m,n+1} \end{align*}(2)
\(n\in\mathbb{N}_{0}\)とする。\begin{align*} \lim_{z\rightarrow0}\left(\frac{\psi^{\left(n\right)}\left(z+\alpha\right)-\psi^{\left(n\right)}\left(z\right)}{\Gamma^{n+1}\left(z\right)}\right) & =\lim_{z\rightarrow0}\left(\frac{\psi^{\left(n\right)}\left(z+\alpha\right)-\psi^{\left(n\right)}\left(z+1\right)+\frac{\left(-1\right)^{n}n!}{z^{n+1}}}{\Gamma^{n+1}\left(z\right)}\right)\\ & =\lim_{z\rightarrow0}\left(\frac{z^{n+1}\left(\psi^{\left(n\right)}\left(z+\alpha\right)-\psi^{\left(n\right)}\left(z+1\right)\right)+\left(-1\right)^{n}n!}{z^{n+1}\Gamma^{n+1}\left(z\right)}\right)\\ & =\lim_{z\rightarrow0}\left(\frac{z^{n+1}\left(\psi^{\left(n\right)}\left(z+\alpha\right)-\psi^{\left(n\right)}\left(z+1\right)\right)+\left(-1\right)^{n}n!}{\Gamma^{n+1}\left(z+1\right)}\right)\\ & =\frac{\left(-1\right)^{n}n!}{\Gamma^{n+1}\left(1\right)}\\ & =\left(-1\right)^{n}n! \end{align*}
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ガンマ関数の相反公式
\[
\Gamma(z)\Gamma(1-z)=\pi\sin^{-1}(\pi z)
\]
1次式の総乗と階乗
\[
\prod_{k=a}^{b}\left(kn+r\right)=n^{b-a+1}\frac{\left(b+\frac{r}{n}\right)!}{\Gamma\left(a+\frac{r}{n}\right)}
\]
ディガンマ関数・ポリガンマ関数の級数表示・テイラー展開と調和数・一般化調和数
\[
\psi\left(z\right)=-\gamma+H_{z-1}
\]
ガンマ関数のハンケル積分表示
\[
\Gamma\left(z\right)=\frac{i}{2\sin\left(\pi z\right)}\int_{C}\left(-\tau\right)^{z-1}e^{-\tau}d\tau
\]