On $\sum_{n=0}^\infty \frac{1}{n^2+bn+c} = \frac{\pi \tan\big(\frac{\pi}2\sqrt{b^2-4c}\big)}{\sqrt{b^2-4c}}+x$ and solvable Galois groups? In this comment, I hastily assumed that, 
$$\sum_{\color{blue}{n=0}}^\infty \frac{1}{n^2+bn+c} = \frac{\pi \tan\big(\frac{\pi}2\sqrt{b^2-4c}\big)}{\sqrt{b^2-4c}}$$ 
In fact, this is valid only for $b=1$ and, as Robert Israel pointed out, for general $b$ is missing a rational term. So for odd $b$ we have,
$$\begin{aligned}
\sum_{n=0}^\infty \frac{1}{n^2+n+c} &= \frac{\pi \tan\big(\frac{\pi}2\sqrt{1^2-4c}\big)}{\sqrt{1^2-4c}}\\
\sum_{n=0}^\infty \frac{1}{n^2+3n+c} &= \frac{\pi \tan\big(\frac{\pi}2\sqrt{3^2-4c}\big)}{\sqrt{3^2-4c}}-\frac1{c-2}\\
\sum_{n=0}^\infty \frac{1}{n^2+5n+c} &= \frac{\pi \tan\big(\frac{\pi}2\sqrt{5^2-4c}\big)}{\sqrt{5^2-4c}}-\frac{2c-10}{(c-4)(c-6)}\\
\sum_{n=0}^\infty \frac{1}{n^2+7n+c} &= \frac{\pi \tan\big(\frac{\pi}2\sqrt{7^2-4c}\big)}{\sqrt{7^2-4c}}-\frac{3c^2-56c+252}{(c-6)(c-10)(c-12)}\end{aligned}$$
But I'm having trouble finding $b=9$.

Questions:



*

*What is the rational term for $b=9$?

*For $b=2m+1$ and $m>0$, is it true that the rational term has form $\displaystyle \frac{P_1(c)}{P_2(c)}$ where $P_1(c)$ is a polynomial of degree $m-1$, while $P_2(c)$ has degree $m$?

*This may be like throwing a curveball, but do the equations $P_1(c)=0$ and $P_2(c)=0$ have solvable Galois groups?


It looks like the correct thing to look at is
$$\sum_{n=-\infty}^\infty\frac1{n^2+bn+c}.$$
This is surely expressible in terms of the digamma function.
Your rational fraction is surely an artefact of insisting on taking a one-sided sum. As an example,
$$\sum_{n=0}^\infty\frac1{n^2+9n+c}
=\sum_{m=-\infty}^{-9}\frac1{(-9-m)^2+9(-9-m)+c}
=\sum_{m=-\infty}^{-9}\frac1{m^2+9m+c}$$
and so
\begin{align}
\sum_{n=0}^\infty\frac1{n^2+9n+c}
&=\frac12\sum_{n=-\infty}^\infty\frac1{n^2+9n+c}-\frac12
\sum_{n=-8}^{-1}\frac1{n^2+9n+c}\\
&=\frac12\sum_{n=-\infty}^\infty\frac1{n^2+9n+c}-
\left(\frac1{c-8}+\frac1{c-14}+\frac1{c-18}+\frac1{c-20}\right)
\end{align}
etc.
