Bessel function of integer order has the property $$ \sum_n J_n(x)e^{in\theta}=e^{ix\sin(\theta)} + C $$
It can be easily derived from the recursive formula $J_{\nu-1}(x)+J_{\nu+1}(x) = \frac{2\nu}{x}J_\nu(x)$ up to a constant C. By taking x to zero, we can get the constant C as zero.
Now I want to make it work for fractional n. Which means $$ \sum_n J_{n+a}(x)e^{i(n+a)\theta}=e^{ix\sin(\theta)} + C(a) $$ where a is the fractional number. It is right because the recursive condition is still valid for the non-integer order of Bessel function.
I'm interested in the case when x is non-zero. However it is hard to determine the value of C(a) since basically, it is an oscillating summation. Is there any helpful advice?