I stepped upon the following sum as part of my research, I haven't been able to solve.
$\sum_{n=1}^{\infty}{e^{i\cdot n\cdot a}}\cdot J_n(z)$
where $a$ and $z$ are both complex numbers, and $J_n$ is the n-th Bessel function.
I tried expanding the Bessel function into a series, this yields the following result
$e^{i\cdot z\cdot \sin(a)}- \sum_{m=0}^{\infty}{{(-z/2\cdot e^{-i\cdot a})^m}/({m!})} \sum_{s=0}^{m}{{(z/2\cdot e^{i\cdot a})^s}/{(s!)}}$
I can also change the summation order, but this gets me nowhere.
it would be great to know if there exists a closed form to this sum.