Referring to a question I posted on MS, I post it here, as I didn't get an answer:

By analogy with the Jacobi–Anger expansion, one expects that $e^{iz\cot(x)}$ has a Fourier expansion of the form : $$e^{iz\cot(\theta)}=\sum_{n=-\infty}^{\infty}\Lambda_{n}(z)e^{in\theta}$$ $\Lambda_{n}(z)$ is given by: $$\Lambda_{n}(z)=\frac{1}{\pi}\int_{-\pi/2}^{\pi/2}e^{iz\cot(\theta)-in\theta}d\theta$$ A simple calculation yields: $$\Lambda_{n}(z)=\frac{2}{n\pi}\sin\left(\frac{\pi n}{2} \right )\sum_{m=0}^{\infty}\frac{z^{m}}{m!}\text{F}_{1}\left(-\frac{n}{2};-m,m;1-\frac{n}{2};1,-1 \right )$$

Where $\text{F}_{1}(\alpha;\beta,\beta^{'};\gamma;x,y)$ is the Appell Hypergeometric Function. Now, I have two questions:

1-For a purely imaginary $z$,$\;\;$ $e^{iz\cot(\theta)}$ has essential singularities at $\theta=\pm n\pi$. How is that reflected in the Fourier expansion?

2-Can we express the infinite sum in terms of other special functions?