Is there any hope to get a closed form for this:
$\sum_{j=0}^{\infty} e^{i \vec{k} \cdot \vec{R_j}} J_{m} (qR_j)$
such that $\vec{R}_j+\vec{a} = \vec{R}_j$, where $\vec{a}$ - constant vector, lattice vector; $m=0,1,2...$? Assume 3D.
It "smells" like a discrete 3D Fourier transform. I am not great with DFT's in general, so any help is highly appreciated.
Also, one can assume $\vec{q}<<\vec{k}$, and use the corresponding expansion for the Bessel function, if it helps.