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

(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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ポリガンマ関数同士の差の極限

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