A series involving digamma function

Question

I am trying to solve the series

$$\sum_{k=1}^\infty\frac{1}{k(k^2+n^2)}$$

The best I got is

$$\frac{\Re\left\{\psi(1+in) \right\}+\gamma)}{n^2}$$

I am not able to simplify it more.

Maybe there is another approach to solve the series. Any idea how ?

You can assume that $n$ is an integer if that simplifies the solution.

Answer 1

In fact, your result is correct and it agrees to Eq.(6.3.17) of Abramovic & Stegun, where you'll find a nice zeta series representation for it. See link http://people.math.sfu.ca/~cbm/aands/page_259.htm