I frequently see expressions for sums of Bessel/Hankel functions over their order. What I have not been able to find is any expressions for sums over their arguments. For example, I am keen to find an expression for
$$ \sum_{n=1}^{\infty}e^{in\alpha}H_{m}^{(1)}(n\beta) $$
where $\alpha$ is real and $\beta>0$. Any useful ideas/references will be greatly appreciated, even if it's just for special cases e.g. $\alpha=0$ or $m=0$.