(1)

\begin{align*} \psi(z) & =\frac{d}{dz}\log\Gamma(z)\\ & =\frac{d}{dz}\log\left(\lim_{n\rightarrow\infty}n^{z}n!P(z,-(n+1))\right)\qquad,\qquad\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}(z+k)^{-1}\right)\\ & =\lim_{n\rightarrow\infty}\frac{d}{dz}\left(\log n^{z}+\log n!-\sum_{k=0}^{n}\log(z+k)\right)\\ & =\lim_{n\rightarrow\infty}\frac{d}{dz}\left(z\log n-\sum_{k=0}^{n}\log(z+k)\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(z) & =\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) \end{align*}

(2)

\(n\in\mathbb{N}\)とする。

\begin{align*} \psi^{(n)}(z) & =\frac{d^{n}}{dz^{n}}\psi(z)\\ & =\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(-1,n)\frac{1}{(z+k)^{n+1}}\right\} \\ & =(-1)^{n+1}n!\sum_{k=0}^{\infty}\frac{1}{(z+k)^{n+1}} \end{align*}

(3)

\(z=1\)周りでテーラー展開すると、

\begin{align*} \psi(1+z) & =\sum_{k=0}^{\infty}\frac{\psi^{(k)}(1)}{k!}z^{k}\\ & =\psi^{(0)}(1)+\sum_{k=1}^{\infty}\frac{\psi^{(k)}(1)}{k!}z^{k}\\ & =-\gamma+H_{0}+\sum_{k=1}^{\infty}\left\{ (-1)^{k+1}\sum_{j=0}^{\infty}\frac{1}{(1+j)^{k+1}}z^{k}\right\} \\ & =-\gamma+\sum_{k=1}^{\infty}\left\{ (-1)^{k+1}\zeta(k+1)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\{ (-1)^{k+1}\zeta(k+1)z^{k}\right\} \right\} \\ & =\sum_{k=1}^{\infty}\left\{ (-1)^{k+1}\zeta(k+1)\frac{d^{n}}{dz^{n}}z^{k}\right\} \\ & =\sum_{k=1}^{\infty}\left\{ (-1)^{k+1}\zeta(k+1)P\left(k,n\right)z^{k-n}\right\} \\ & =\sum_{k=n}^{\infty}\left\{ (-1)^{k+1}\zeta(k+1)P\left(k,n\right)z^{k-n}\right\} \\ & =\left(-1\right)^{n+1}\sum_{k=0}^{\infty}(-1)^{k}\zeta(n+k+1)P\left(n+k,n\right)z^{k} \end{align*}