This can be expressed in terms of elementary functions, if $p/2$ is an integer. Suppose, for example that it is a positive integer. Then your sum is
$$S:=\sum_{m=p/2+1}^\infty\frac{1}{m^2+c}=\frac{1}{2}\sum_{|m|&gt;p/2}\frac{1}{m^2+c},\quad c=q-p^2/4,$$
where summation in the last sum is over positive and negative integers. The sum in the right hand side differs by finitely many summands from the sum
$$\sum_{-\infty}^\infty\frac{1}{m^2+c},$$
which you know how to compute. The simplest way to do this, by the way, is by the residue theorem: integrate 
$$\frac{\pi\cot\pi z}{z^2+c}$$ over appropriate expanding contours.
The answer will be slightly different, depending on whether $\sqrt{-c}$
is an integer or not. Thus you obtain a closed form
answer in elementary functions. The case when $p/2$ is a negative integer it treated similarly, by first dropping finitely many terms of your sum, and then adding them back. 

If $p/2$ is not an integer, I afraid that you have to use Gamma function, as suggested in the comments, and there is no
way around this.

Series of the inverse quadratic trinomial
I need to extend selection of external and Internal Link and provide a image selection.

Please see the snapshot in the below :


Here the above snapshot allows you to add properties for External Link.In the same popup can we add a field Called Image(as shown in screenshot) which will allow user to select images from the media library??

Thanks,
Suhas
