My main question: Can we obtain the exact solutions from the following equation?
$$ \sum_{k=1}^{n}\cfrac{1}{k-x-1}=0 $$
Notation: This problem was reached from the digamma function $\psi$ as follows:
$$ \sum_{k=1}^{n}\cfrac{1}{k-x-1}=\psi(n-x)-\psi(-x) $$
Generally, any polynomial equations can be solved exactly until four order as the quartic equation. Nevertheless, in high order case in this problem $(n\geq 6)$, we can also obtain the exact solutions. Here is a table which is described since beginning of $n=2$:
$$ \begin{aligned} \text{equation} && && \text{solutions} \\ \psi(2-x)=\psi(-x) && \cfrac{1}{2} \\ \psi(3-x)=\psi(-x) && 1\pm\cfrac{1}{\sqrt{3}} \\ \psi(4-x)=\psi(-x) && \cfrac{3}{2} && \cfrac{3\pm\sqrt{5}}{2} \\ \psi(5-x)=\psi(-x) && 2\pm\sqrt{\cfrac{15\pm\sqrt{145}}{10}} \\ \psi(6-x)=\psi(-x) && \cfrac{5}{2} && \cfrac{5}{2}\pm\cfrac{\sqrt{3(35\pm 8\sqrt{7})}}{6} \\ \vdots && \vdots && \vdots \end{aligned} $$
Furthermore, suppose $n=8$ and $n=9$ cases (Of course, other cases are true) and these algebraic solutions are existed:
solutions of $n=8$ by Wolfram Alpha
solutions of $n=9$ by Wolfram Alpha
Additionally, these approximate solutions are estimated on each equation and the diagram is shown below:
Then, horizontal axis is $x$ and the vertical axis is $n$. Expressions of periodic functions can be expected because we can see a periodical pattern from diagrams. In fact, these solutions including nested rationals have deep relations with trigonometric functions. Example of expressions are shown below (if you want to know further, we can see this site and this paper.):
$$ \begin{aligned} \sin\cfrac{\pi}{10}=&\cfrac{\sqrt{5}-1}{4} && \cos\cfrac{\pi}{10}=&\cfrac{\sqrt{2(5+\sqrt{5})}}{4} && \tan\cfrac{\pi}{10}=&\cfrac{\sqrt{ 5(5-2\sqrt{5}) }}{5} \\ \sin\cfrac{\pi}{5}=&\cfrac{\sqrt{2(5-\sqrt{5})}}{4} && \cos\cfrac{\pi}{5}=&\cfrac{\sqrt{5}+1}{4} && \tan\cfrac{\pi}{5}=&\sqrt{5-2\sqrt{5}} \end{aligned} $$
And reflection formula may be useful knowledge:
$$ \psi(1-x)-\psi(x)=\pi\cot(\pi x) $$
As the above description, we can guess that these solutions can be converted into trigonometric functions. I am very glad that someone indicates any advises or references.