Are there any equivalent representations of the following (real valued) sum, in particular that are suitable for evaluation as $z\rightarrow0$ ?
$$ S=\sum\_{k=-\infty}^\infty \frac{i^k(z-2ik)}{(\rho^2+(z-2ik)^2)^{3/2}} $$


I am aware that $S$ resembles a coulomb force sum which can be rearranged using Lekner summation into a seires of Bessel functions, but the factor $i^k$ seems to prohibit this transformation.



Alternating 1D lattice sum
For $\rho$ equal to an even integer the sum $S$ diverges as $1/z^{3/2}$ when $z\rightarrow 0$. For $\rho$ unequal to an even integer and $z>0$, one has
$$S\_0=\lim\_{z\rightarrow 0}\sum\_{k=-\infty}^\infty \frac{i^k(z-2ik)}{(\rho^2+(z-2ik)^2)^{3/2}}=-4\,\Re\sum\_{k=1}^\infty\frac{ i^k k}{\left(4 k^2-\rho^2\right)^{3/2}}.$$


