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.
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 $$S_0=\lim_{z\rightarrow 0}\sum_{k=-\infty}^\infty \frac{i^k(z-2ik)}{(\rho^2+(z-2ik)^2)^{3/2}}=-4\,\Re\sum_{k=1}^\infty\frac{ i^k k}{\left(4 k^2-\rho^2\right)^{3/2}}.$$
$S$ may be expressed as a series of Bessel functions:
$$S=-\pi^2\sum_{n=1}^\infty \left(n-1/4\right) \exp[-(n-1/4)\pi z] J_0[(n-1/4)\pi\rho],$$
however this form diverges for $z=0$.
