ポリガンマ関数の特殊値(1/2,1/3,2/3,1/4,3/4,1/6,5/6)

$\begin{array}{rl}\frac{1}{n}\sum _{k=0}^{n-1}\psi \left(z+\frac{k}{n}\right)& =\frac{1}{n}\sum _{k=0}^{n-1}\underset{N\to \mathrm{\infty }}{lim}\left(\ln N-\sum _{l=0}^N\frac{1}{z+\frac{k}{n}+l}\right)\\ & =\underset{N\to \mathrm{\infty }}{lim}\left(\ln N-\sum _{k=0}^{n-1}\sum _{l=0}^N\frac{1}{nz+nl+k}\right)\\ & =\underset{N\to \mathrm{\infty }}{lim}\left(\ln N-\sum _{j=0}^{Nn+n-1}\frac{1}{nz+j}\right)=\underset{N\to \mathrm{\infty }}{lim}\left(\ln N-\sum _{j=0}^{Nn}\frac{1}{nz+j}\right)\\ & =\underset{N\to \mathrm{\infty }}{lim}\left(\ln \left(Nn\right)-\sum _{j=0}^{Nn}\frac{1}{nz+j}\right)-\ln n=\psi (nz)-\ln n\\ n\psi (nz)& =n\ln n+\sum _{k=0}^{n-1}\psi \left(z+\frac{k}{n}\right)\end{array}$
これを$m$階微分して定理を得ます。

定理1に$z=\frac{1}{2},n=2$を代入すると、
$2^{m+1}{\psi }^{(m)}(1)={\delta }_{m,0}2\ln 2+{\psi }^{(m)}\left({\displaystyle \frac{1}{2}}\right)+{\psi }^{(m)}(1)$

$\begin{array}{rl}& \mathsf{\text{定理1に}}z=\frac{1}{4},n=2\mathsf{\text{を代入すると、}}\\ {\psi }^{(m)}\left(\frac{1}{4}\right)+{\psi }^{(m)}\left(\frac{3}{4}\right)& =2^{m+1}{\psi }^{(m)}\left(\frac{1}{2}\right)-{\delta }_{m,0}2\ln 2\\ & =2^{m+1}\left[\left(2^{m+1}-1\right){\psi }^{(m)}(1)-{\delta }_{m,0}2\ln 2\right]-{\delta }_{m,0}2\ln 2\\ & =2^{m+1}\left(2^{m+1}-1\right){\psi }^{(m)}(1)-6{\delta }_{m,0}\ln 2\\ \\ {\psi }^{(m)}\left(\frac{1}{4}\right)-{\psi }^{(m)}\left(\frac{3}{4}\right)& =(-4{)}^{m+1}m!\beta (m+1)\\ \\ {\psi }^{(m)}\left(\frac{2\pm 1}{4}\right)& =\frac{(\text{上})\mp (\text{下})}{2}\end{array}$

$\begin{array}{rl}& \mathsf{\text{定理1に}}z=\frac{1}{3},n=3\mathsf{\text{を代入すると、}}\\ 3^{2m+1}{\psi }^{(2m)}(1)& ={\delta }_{m,0}3\ln 3+{\psi }^{(2m)}\left(\frac{1}{3}\right)+{\psi }^{(2m)}\left(\frac{2}{3}\right)+{\psi }^{(2m)}(1)\\ {\psi }^{(2m)}\left(\frac{1}{3}\right)+{\psi }^{(2m)}\left(\frac{2}{3}\right)& =\left(3^{2m+1}-1\right){\psi }^{(m)}(1)-{\delta }_{m,0}3\ln 3\\ & \mathsf{\text{相反公式より、}}\\ {\psi }^{(2m)}\left(\frac{1}{3}\right)-{\psi }^{(2m)}\left(\frac{2}{3}\right)& =-\pi \cdot {(\cot \left(\pi x\right){)}^{(2m)}|}_{x=\frac{1}{3}}\\ \\ {\psi }^{(2m)}\left(\frac{3\pm 1}{6}\right)& =\frac{(\text{上})\mp (\text{下})}{2}\end{array}$

$\begin{array}{rl}& \mathsf{\text{定理1の}}n=2\mathsf{\text{の場合において、}}\\ (\text{左辺})& +{\psi }^{(2m)}\left(\frac{3\mp 1}{6}\right)=2^{2m+1}{\psi }^{(2m)}\left(\frac{3\pm 1}{6}\right)-{\delta }_{m,0}2\ln 2\text{※}min(3\pm 2,3\mp 1)=\frac{3\pm 1}{2}\\ (\text{左辺})& =2^{2m+1}{\psi }^{(2m)}\left(\frac{3\pm 1}{6}\right)-{\psi }^{(2m)}\left(\frac{3\mp 1}{6}\right)-{\delta }_{m,0}2\ln 2\\ & \begin{array}{rl}=& 2^{2m}\left[\left(3^{2m+1}-1\right){\psi }^{(2m)}(1)\pm \pi \cdot {(\cot \left(\pi x\right){)}^{(2m)}|}_{x=\frac{1}{3}}-{\delta }_{m,0}3\ln 3\right]\\ & -\frac{1}{2}\left[\left(3^{2m+1}-1\right){\psi }^{(2m)}(1)\mp \pi \cdot {(\cot \left(\pi x\right){)}^{(2m)}|}_{x=\frac{1}{3}}-{\delta }_{m,0}3\ln 3\right]\\ & -{\delta }_{m,0}2\ln 2\end{array}\\ & =(\text{右辺})\end{array}$