Closed form with of a series Mathematica I've these two series, and I would like a closed form:
$$ \sum_{k=-\infty}^{\infty} \frac{x+kx_0-h}{|x+kx_0|^3}$$
$$ 3\sum_{k=-\infty}^{\infty} \frac{(x+kx_0-h)(x+kx_0)^2}{|x+kx_0|^5} $$
Mathematica gives me these closed forms ($A=x_0$, $B=h$): 
Sum[(x + k*A - B)/((abs (x + k*A))^3), {k, -Infinity, Infinity}]


  =   (1/(2 A^3 abs^3))(2 A PolyGamma[1, x/A] + 2 A PolyGamma[1, 1 - x/A] + 
     B PolyGamma[2, x/A] - B PolyGamma[2, 1 - x/A])


Sum[((x + k*A - B)*(x + k*A)^2)/(abs (x + k*A)^5), {k, -Infinity, Infinity }]


=  (1/(2 A^3 abs))(2 A PolyGamma[1, x/A] + 2 A PolyGamma[1, 1 - x/A] + B  
   PolyGamma[2, x/A] - B PolyGamma[2, 1 - x/A])


Now, first of all, when it writes $abs^3$ this should mean $|x|^3$ or $|x-kx_0|^3$?
And at last, does this make sense? Is there an analytical way to reach this (or a better) conclusion?
Thanks!
 A: If $0 < x < x_0$, $x + k x_0 > 0$ for $k \ge 0$ and $< 0$ for $k \le -1$, so your first sum is
$$ - \sum_{k=-\infty}^{-1} \frac{x + k x_0 - h}{(x + k x_0)^3} +   \sum_{k=0}^{\infty} \frac{x + k x_0 - h}{(x + k x_0)^3}$$
which Maple evaluates as 
$$ \frac{ h\Psi \left( 2,{\dfrac {x}{{ x_0}}} \right) +2 { x_0}\ \Psi
 \left( 1,{\dfrac {x}{ x_0}}\right) { x_0}}{2\, { x_0}^3}+ \frac{ h\Psi \left( 2,1-\dfrac{x}{ x_0}
 \right) -2\,{x_0}\,\Psi \left( 1,1-\dfrac{x}{x_0}
 \right)  } {2\, {x_0}^3}
$$
The second sum is just $3$ times the first if $x$ and $x_0$ are real, because $(x + k x_0)^2 = \left| x + k x_0 \right|^2$.
EDIT: If $x_0 > 0$ and $s$ is the fractional part of $x/x_0$ (so $0 < s < 1$ and $x/x_0 - s$ is an integer) then $x + k x_0 > 0$ for $k \ge n = s - x/x_0$ and $<0$ for $k \le n-1$, your first sum is
$$  - \sum_{k=-\infty}^{n-1} \frac{x + k x_0 - h}{(x + k x_0)^3} +   \sum_{k=n}^{\infty} \frac{x + k x_0 - h}{(x + k x_0)^3}$$
which Maple evaluates as 
$$ \dfrac{2 \Psi(1,s)}{ x_0^2} + \dfrac{h \Psi(2,s)}{x_0^3} + \dfrac{\pi^3 h \cot(\pi s) \csc(\pi s)^2}{x_0^3} - \dfrac{\pi^2 \csc(\pi s)^2}{x_0^2}
$$
