(1)

\begin{align*} \psi\left(z+1\right) & =\frac{d}{d\left(z+1\right)}\log\Gamma\left(z+1\right)\\ & =\frac{dz}{d\left(z+1\right)}\frac{d}{dz}\log\left(z\Gamma\left(z\right)\right)\\ & =\frac{d}{dz}\left(\log z+\log\Gamma\left(z\right)\right)\\ & =\frac{1}{z}+\psi\left(z\right) \end{align*}

(1)-2

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

(2)

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

(2)-2

\begin{align*} \psi^{\left(n\right)}(z+1) & =\frac{d^{n}}{dz^{n}}\psi(z+1)\\ & =\frac{d^{n+1}}{dz^{n+1}}\log\Gamma(z+1)\\ & =\frac{d^{n+1}}{dz^{n+1}}\log\left(z\Gamma(z)\right)\\ & =\frac{d^{n+1}}{dz^{n+1}}\left(\log\Gamma(z)+\log z\right)\\ & =\frac{d^{n}}{dz^{n}}\psi(z)+\frac{d^{n}}{dz^{n}}\frac{1}{z}\\ & =\psi^{\left(n\right)}(z)+\frac{\left(-1\right)^{n}n!}{z^{n+1}} \end{align*}

(3)

\begin{align*} \psi(1) & =-\gamma-\sum_{k=0}^{\infty}\left(\frac{1}{1+k}-\frac{1}{k+1}\right)\\ & =-\gamma \end{align*}

(4)

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

(5)

\begin{align*} \psi(m) & =\sum_{j=1}^{m-1}\left(\psi(j+1)-\psi(j)\right)+\psi(1)\\ & =\sum_{j=1}^{m-1}\frac{1}{j}+\psi(1)\\ & =H_{m-1}-\gamma \end{align*}

(6)

\begin{align*} \psi^{\left(n\right)}\left(z+m\right) & =\psi^{\left(n\right)}\left(z+1\right)+\sum_{k=1}^{m-1}\left(\psi^{\left(n\right)}\left(z+k+1\right)-\psi^{\left(n\right)}\left(z+k\right)\right)\\ & =\psi^{\left(n\right)}\left(z+1\right)+\sum_{k=1}^{m-1}\frac{\left(-1\right)^{n}n!}{\left(z+k\right)^{n+1}}\\ & =\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}} \end{align*}

(7)

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

(8)

\begin{align*} \psi\left(\frac{1}{2}\right) & =-\gamma-\lim_{n\rightarrow\infty}\sum_{k=0}^{n}\left(\frac{1}{\frac{1}{2}+k}-\frac{1}{k+1}\right)\\ & =-\gamma-\lim_{n\rightarrow\infty}\sum_{k=0}^{n}\left(\frac{2}{1+2k}-\frac{1}{k+1}\right)\\ & =-\gamma-2\lim_{n\rightarrow\infty}\sum_{k=0}^{n}\left(\frac{1}{1+2k}-\frac{1}{2(k+1)}\right)\\ & =-\gamma-2\lim_{n\rightarrow\infty}\sum_{k=0}^{n}\left(\frac{(-1)^{2k+1+1}}{2k+1}+\frac{(-1)^{2(k+1)+1}}{2(k+1)}\right)\\ & =-\gamma-2\lim_{n\rightarrow\infty}\sum_{k=1}^{2n+2}\left(\frac{(-1)^{k+1}}{k}\right)\\ & =-\gamma-2\log2 \end{align*}

(9)

\begin{align*} \psi^{(n)}\left(\frac{1}{2}\right) & =(-1)^{n+1}n!\sum_{k=0}^{\infty}\frac{1}{\left(\frac{1}{2}+k\right)^{n+1}}\\ & =(-1)^{n+1}n!2^{n+1}\sum_{k=0}^{\infty}\frac{1}{\left(2k+1\right)^{n+1}}\\ & =(-1)^{n+1}n!2^{n+1}\left(\zeta\left(n+1\right)-\frac{\zeta\left(n+1\right)}{2^{n+1}}\right)\\ & =-(-1)^{n}n!\left(2^{n+1}-1\right)\left(\zeta\left(n+1\right)\right) \end{align*}

(10)

\(n\in\mathbb{N}\)のとき、

\begin{align*} \psi\left(m-\frac{1}{2}\right) & =\sum_{j=0}^{m-2}\left(\psi\left(j+1+\frac{1}{2}\right)-\psi\left(j+\frac{1}{2}\right)\right)+\psi(\frac{1}{2})\\ & =\sum_{j=0}^{m-2}\frac{1}{j+\frac{1}{2}}+\psi\left(\frac{1}{2}\right)\\ & =2\sum_{j=0}^{m-2}\frac{1}{2j+1}+\psi\left(\frac{1}{2}\right)\\ & =2\left(\sum_{j=1}^{2m-2}\frac{1}{j}-\sum_{j=1}^{m-1}\frac{1}{2j}\right)+\psi\left(\frac{1}{2}\right)\\ & =2\left(\sum_{j=1}^{m-1}\frac{1}{2j}+\sum_{j=1}^{m-1}\frac{1}{2j-1}-\sum_{j=1}^{m-1}\frac{1}{2j}\right)+\psi\left(\frac{1}{2}\right)\\ & =H_{m-1}-2\sum_{j=1}^{2m-2}\frac{(-1)^{j}}{j}-\gamma-2\log2 \end{align*}