(1)

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

これより、
\[ \psi(1-z)-\psi(z)=\pi\tan^{-1}(\pi z) \]

(2)

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

これより、

\[ \left(-1\right)^{n}\psi^{\left(n\right)}\left(1-z\right)-\psi^{\left(n\right)}\left(z\right)=\pi\frac{d^{n}}{dz^{n}}\tan^{-1}\left(\pi z\right) \]

(3)

\(n\in\mathbb{N}\)のとき、
\begin{align*} \lim_{z\rightarrow n}\frac{\psi(1-z)}{\Gamma(1-z)} & =\lim_{z\rightarrow n}\frac{\Gamma(z)}{\pi\sin^{-1}(\pi z)}\left\{ \psi(z)+\pi\tan^{-1}(\pi z)\right\} \\ & =\lim_{z\rightarrow n}\frac{\Gamma(z)\sin(\pi z)}{\pi}\left\{ H_{n-1}-\gamma+\pi\frac{\cos(\pi z)}{\sin(\pi z)}\right\} \\ & =\cos(\pi n)\Gamma(n)\\ & =(-1)^{n}(n-1)! \end{align*}