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
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
\begin{align*} \psi\left(z\right) & =\lim_{n\rightarrow\infty}\left(\log n-\sum_{k=0}^{n}\frac{1}{z+k}\right)\\ & =-\gamma-\sum_{k=0}^{\infty}\left(\frac{1}{z+k}-\frac{1}{k+1}\right)\\ & =-\gamma+H_{z-1} \end{align*}
\(n\in\mathbb{N}\)とする。
\begin{align*} \psi^{\left(n\right)}\left(z\right) & =\left(-1\right)^{n+1}n!\sum_{k=0}^{\infty}\frac{1}{\left(z+k\right)^{n+1}}\\ & =\left(-1\right)^{n+1}n!\zeta\left(n+1,z\right)\\ & =\left(-1\right)^{n+1}n!\left(\zeta\left(n+1\right)-H_{z-1,n+1}\right) \end{align*}
\[ \psi^{\left(n\right)}\left(z+1\right)=\left(-1\right)^{n+1}\sum_{k=0}^{\infty}(\left(-1\right)\zeta\left(n+k+1\right)P\left(n+k,n\right)z^{k} \]
\(\zeta\left(n,z\right)\)はフルヴィッツ・ゼータ関数関数
\(H_{x}\)は調和数
\(H_{x,n}\)は一般化調和数
(1)ディガンマ関数の級数表示
\(z\in\mathbb{C}\setminus\mathbb{N}_{0}^{-}\)とする。\begin{align*} \psi\left(z\right) & =\lim_{n\rightarrow\infty}\left(\log n-\sum_{k=0}^{n}\frac{1}{z+k}\right)\\ & =-\gamma-\sum_{k=0}^{\infty}\left(\frac{1}{z+k}-\frac{1}{k+1}\right)\\ & =-\gamma+H_{z-1} \end{align*}
(2)ポリガンマ関数の級数表示
\(z\in\mathbb{C}\setminus\mathbb{N}_{0}^{-}\)とする。\(n\in\mathbb{N}\)とする。
\begin{align*} \psi^{\left(n\right)}\left(z\right) & =\left(-1\right)^{n+1}n!\sum_{k=0}^{\infty}\frac{1}{\left(z+k\right)^{n+1}}\\ & =\left(-1\right)^{n+1}n!\zeta\left(n+1,z\right)\\ & =\left(-1\right)^{n+1}n!\left(\zeta\left(n+1\right)-H_{z-1,n+1}\right) \end{align*}
(3)ディガンマ関数のテイラー展開
\[ \psi\left(z+1\right)=-\gamma+\sum_{k=1}^{\infty}\left\{ (\left(-1\right)\zeta\left(k+1\right)z^{k}\right\} \](4)ポリガンマ関数のテイラー展開
\(n\in\mathbb{N}\)とする。\[ \psi^{\left(n\right)}\left(z+1\right)=\left(-1\right)^{n+1}\sum_{k=0}^{\infty}(\left(-1\right)\zeta\left(n+k+1\right)P\left(n+k,n\right)z^{k} \]
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\(\zeta\left(n\right)\)はリーマン・ゼータ関数\(\zeta\left(n,z\right)\)はフルヴィッツ・ゼータ関数関数
\(H_{x}\)は調和数
\(H_{x,n}\)は一般化調和数
(1)
\begin{align*} \psi\left(z\right) & =\frac{d}{dz}\log\Gamma\left(z\right)\\ & =\frac{d}{dz}\log\left(\lim_{n\rightarrow\infty}n^{z}n!P\left(z,-\left(n+1\right)\right)\right)\cmt{\text{ガウスのガンマ関数無限乗積表示}}\\ & =\frac{d}{dz}\log\left(\lim_{n\rightarrow\infty}n^{z}n!\prod_{k=0}^{n}\left(z+k\right)^{-1}\right)\\ & =\lim_{n\rightarrow\infty}\frac{d}{dz}\left(\log n^{z}+\log n!-\sum_{k=0}^{n}\log\left(z+k\right)\right)\\ & =\lim_{n\rightarrow\infty}\frac{d}{dz}\left(z\log n-\sum_{k=0}^{n}\log\left(z+k\right)\right)\\ & =\lim_{n\rightarrow\infty}\left(\log n-\sum_{k=0}^{n}\frac{1}{z+k}\right) \end{align*} 更に計算を進めると、\begin{align*} \psi\left(z\right) & =\lim_{n\rightarrow\infty}\left\{ \log n-\sum_{k=0}^{n}\frac{1}{k+1}-\sum_{k=0}^{n}\left(\frac{1}{z+k}-\frac{1}{k+1}\right)\right\} \\ & =-\gamma-\sum_{k=0}^{\infty}\left(\frac{1}{z+k}-\frac{1}{k+1}\right)\\ & =-\gamma-\sum_{k=1}^{\infty}\left(\frac{1}{z-1+k}-\frac{1}{k}\right)\\ & =-\gamma+\sum_{k=1}^{\infty}\left(\frac{1}{k}-\frac{1}{k+z-1}\right)\\ & =-\gamma+H_{z-1} \end{align*}
(2)
\(n\in\mathbb{N}\)とする。\begin{align*} \psi^{(n)}\left(z\right) & =\frac{d^{n}}{dz^{n}}\psi\left(z\right)\\ & =\frac{d^{n}}{dz^{n}}\left\{ -\gamma-\sum_{k=0}^{\infty}\left(\frac{1}{z+k}-\frac{1}{k+1}\right)\right\} \\ & =-\left\{ \sum_{k=0}^{\infty}\frac{d^{n}}{dz^{n}}\frac{1}{z+k}\right\} \\ & =-\left\{ \sum_{k=0}^{\infty}P\left(-1,n\right)\frac{1}{\left(z+k\right)^{n+1}}\right\} \\ & =\left(-1\right)^{n+1}n!\sum_{k=0}^{\infty}\frac{1}{\left(z+k\right)^{n+1}} \end{align*}
(2)-2
\begin{align*} \psi^{\left(n\right)}\left(z\right) & =\lim_{m\rightarrow\infty}\left\{ \sum_{k=0}^{m}\left(\psi^{\left(n\right)}\left(z+k\right)-\psi^{\left(n\right)}\left(z+1+k\right)\right)+\psi^{\left(n\right)}\left(z+1+m\right)\right\} \\ & =\lim_{m\rightarrow\infty}\left\{ \sum_{k=0}^{m}\left(-\frac{\left(-1\right)^{n}n!}{\left(z+k\right)^{n+1}}\right)+\psi^{\left(n\right)}\left(z+1+m\right)\right\} \\ & =\left(-1\right)^{n+1}m!\sum_{k=0}^{\infty}\frac{1}{\left(z+k\right)^{n+1}}\\ & =\left(-1\right)^{n+1}n!\zeta\left(n+1,z\right) \end{align*}(3)
\(z=1\)周りでテーラー展開すると、\begin{align*} \psi\left(1+z\right) & =\sum_{k=0}^{\infty}\frac{\psi^{(k)}\left(1\right)}{k!}z^{k}\\ & =\psi^{(0)}\left(1\right)+\sum_{k=1}^{\infty}\frac{\psi^{(k)}\left(1\right)}{k!}z^{k}\\ & =-\gamma+H_{0}+\sum_{k=1}^{\infty}\left\{ \left(-1\right)^{k+1}\sum_{j=0}^{\infty}\frac{1}{\left(1+j\right)^{k+1}}z^{k}\right\} \\ & =-\gamma+\sum_{k=1}^{\infty}\left\{ \left(-1\right)^{k+1}\zeta\left(k+1\right)z^{k}\right\} \end{align*}
(4)
\begin{align*} \psi^{\left(n\right)}\left(z+1\right) & =\left[\frac{d^{n}}{dz^{n}}\psi\left(z\right)\right]_{z\rightarrow z+1}\\ & =\left(\frac{dz}{d\left(z+1\right)}\frac{d}{dz}\right)^{n}\psi\left(z+1\right)\\ & =\frac{d^{n}}{dz^{n}}\psi\left(z+1\right)\\ & =\frac{d^{n}}{dz^{n}}\left\{ -\gamma+\sum_{k=1}^{\infty}\left\{ \left(-1\right)^{k+1}\zeta\left(k+1\right)z^{k}\right\} \right\} \\ & =\sum_{k=1}^{\infty}\left\{ \left(-1\right)^{k+1}\zeta\left(k+1\right)\frac{d^{n}}{dz^{n}}z^{k}\right\} \\ & =\sum_{k=1}^{\infty}\left\{ \left(-1\right)^{k+1}\zeta\left(k+1\right)P\left(k,n\right)z^{k-n}\right\} \\ & =\sum_{k=n}^{\infty}\left\{ \left(-1\right)^{k+1}\zeta\left(k+1\right)P\left(k,n\right)z^{k-n}\right\} \\ & =\left(-1\right)^{n+1}\sum_{k=0}^{\infty}\left(-1\right)^{k}\zeta\left(n+k+1\right)P\left(n+k,n\right)z^{k} \end{align*}(4)-2
\(z=1\)周りでテーラー展開すると、\begin{align*} \psi^{\left(n\right)}\left(1+z\right) & =\sum_{k=0}^{\infty}\frac{\psi^{(n+k)}\left(1\right)}{k!}z^{k}\\ & =\sum_{k=0}^{\infty}\frac{\left(-1\right)^{n+k+1}\left(n+k\right)!\zeta\left(n+k+1,1\right)}{k!}z^{k}\\ & =\sum_{k=0}^{\infty}\left(-1\right)^{n+k+1}\frac{\left(n+k\right)!}{k!}\zeta\left(n+k+1\right)z^{k} \end{align*}
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ガンマ関数の絶対収束条件
ガンマ関数$\Gamma\left(z\right)$は$\Re\left(z\right)>0$で絶対収束
ガンマ関数を含む極限
\[
\lim_{n\rightarrow\infty}\sqrt{n}\frac{\Gamma\left(n\right)}{\Gamma\left(n+\frac{1}{2}\right)}=1
\]
ガンマ関数の微分
\[
\frac{d}{dz}\Gamma(z)=\Gamma(z)\psi(z)
\]
偶数と奇数の2重階乗
\[
\left(2n+1\right)!!=2^{n+1}\frac{\left(n+\frac{1}{2}\right)!}{\Gamma\left(\frac{1}{2}\right)}
\]