Editing Digamma function (section)

==Some finite sums involving the digamma function==
There are numerous finite summation formulas for the digamma function. Basic summation formulas, such as

:<math>\sum_{r=1}^m \psi\left(\frac{r}{m}\right)=-m(\gamma+\ln m),</math>
:<math>\sum_{r=1}^m \psi\left(\frac{r}{m}\right)\cdot\exp\dfrac{2\pi rki}{m} = m\ln \left(1-\exp\frac{2\pi ki}{m}\right), \qquad k\in\Z,\quad m\in\N,\ k\ne m</math>
:<math>\sum_{r=1}^{m-1} \psi\left(\frac{r}{m}\right)\cdot\cos\dfrac{2\pi rk}{m} = m \ln \left(2\sin\frac{k\pi}{m}\right)+\gamma, \qquad k=1, 2,\ldots, m-1 </math>
: <math>\sum_{r=1}^{m-1}\psi \left(\frac{r}{m}\right) \cdot\sin\frac{2\pi rk}{m} =\frac{\pi}{2} (2k-m), \qquad  k=1, 2,\ldots, m-1 </math>

are due to Gauss.<ref>R. Campbell. ''Les intégrales eulériennes et leurs applications'', Dunod, Paris, 1966.</ref><ref>H.M. Srivastava and J. Choi. ''Series Associated with the Zeta and Related Functions'', Kluwer Academic Publishers, the Netherlands, 2001.</ref> More complicated formulas, such as

: <math>\sum_{r=0}^{m-1} \psi \left(\frac{2r+1}{2m}\right)\cdot\cos\frac{(2r+1)k\pi }{m} = m\ln\left(\tan\frac{\pi k}{2m}\right) ,\qquad k=1, 2,\ldots, m-1</math>
: <math>\sum_{r=0}^{m-1} \psi \left(\frac{2r+1}{2m}\right)\cdot\sin\dfrac{(2r+1)k\pi }{m} = -\frac{\pi m}{2}, \qquad k=1, 2,\ldots, m-1</math>
:<math>\sum_{r=1}^{m-1} \psi\left(\frac{r}{m}\right)\cdot\cot\frac{\pi r}{m}= -\frac{\pi(m-1)(m-2)}{6}</math>
:<math>\sum_{r=1}^{m-1}\psi \left(\frac{r}{m}\right)\cdot \frac{r}{m}=-\frac{\gamma}{2}(m-1)-\frac{m}{2}\ln m -\frac{\pi}{2}\sum_{r=1}^{m-1} \frac{r}{m}\cdot\cot\frac{\pi r}{m} </math>
:<math>\sum_{r=1}^{m-1}\psi \left(\frac{r}{m}\right) \cdot\cos\dfrac{(2\ell+1)\pi r}{m}= -\frac{\pi}{m}\sum_{r=1}^{m-1} \frac{r \cdot\sin\dfrac{2\pi r}{m}}{\cos\dfrac{2\pi r}{m} -\cos\dfrac{(2\ell+1)\pi }{m} }, \qquad \ell\in\mathbb{Z} </math>
:<math>\sum_{r=1}^{m-1}\psi \left(\frac{r}{m}\right) \cdot\sin\dfrac{(2\ell+1)\pi r}{m}=-(\gamma+\ln2m)\cot\frac{(2\ell+1)\pi}{2m} + \sin\dfrac{(2\ell+1)\pi }{m}\sum_{r=1}^{m-1} \frac{\ln\sin\dfrac{\pi r}{m}} {\cos\dfrac{2\pi r}{m} -\cos\dfrac{(2\ell+1)\pi }{m} } , \qquad \ell\in\mathbb{Z}</math>
:<math>\sum_{r=1}^{m-1} \psi^2\left(\frac{r}{m}\right)= (m-1)\gamma^2 + m(2\gamma+\ln4m)\ln{m} -m(m-1)\ln^2 2 +\frac{\pi^2 (m^2-3m+2)}{12} +m\sum_{\ell=1}^{ m-1 } \ln^2 \sin\frac{\pi\ell}{m}</math>

are due to works of certain modern authors (see e.g. Appendix B in Blagouchine (2014)<ref name="iaroslav_07">{{cite journal|doi=10.1016/j.jnt.2014.08.009 |first=Iaroslav V. |last=Blagouchine |title=A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations |journal=Journal of Number Theory |volume=148 |pages=537–592 |date=2014 |arxiv=1401.3724}}</ref>).

We also have <ref>{{cite book |title=Classical topi s in complex function theorey |pages=46}}</ref>
:<math> 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{k-1}-\gamma=\frac{1}{k}\sum_{n=0}^{k-1}\psi\left(1+\frac{n}{k}\right), k=2,3, ...</math>