In Jacobi-Anger expansion, $$e^{\iota z \sin(\theta)}$$ can be written as: $$e^{\iota z \sin(\theta)} = \sum_{n=-\infty}^{\infty} J_n(z)e^{\iota n \theta}$$
where $J_n(z)$ is the Bessel function of first kind with argument $z$ and order $\nu$. I have an expansion that looks like something similar but I am not sure how it could be condensed. Here's the summation I have:
$$\sum_{n=-\infty}^{\infty} (-1)^n J_n(z)\frac{e^{\iota n \theta}}{A + \iota n B}$$
Any recommendations how to recast it as an exponential of a trigonometric function? It looks very similar to the jacobi-Anger expansion. I tried to expand the summation and see if a trend emerges but no success so far.
