(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*}