I'm interested in finding the analytical continuation of the function $$F(z) = \sum_{n=-\infty}^\infty \frac{\csc\left(\pi z(n+\frac{1}{2})\right)}{\Gamma\!\left(1+ z(n+\frac{1}{2})\right)}$$ The series as defined only seems to converge for purely imaginary values of $z$. My initial thought was to use the integral representation of the reciprocal gamma function, however, the residues of $\csc$ seem to make this much more difficult. I think there might be some way to get Poisson summation to work here, but I couldn't figure out how.
Context:
It's possible for a single formula to converge to two different holomorphic functions in two disconnected domains. For instance,
$$ \sum_{n=0}^\infty \frac{x^{2^{n}}}{1-x^{2^{n+1}}} = \begin{cases} \frac{x}{1-x} & |x|<1 \\ \frac{x}{1-x} + 1 & |x|>1\end{cases}$$
A natural question to ask is "what is the relationship between the different functions represented by the same formula?" Here is one particular case to consider. Take the Lambert-like series
$$f(x) = \sum_{n=0}^\infty (-1)^n a_n \frac{x^n}{1+x^n} $$
Let $F(x) = \sum_{n=0}^\infty a_n x^n$. A heuristic computation says that
$$f(x) - \sum_{k=0}^\infty (-1)^k F(-x^k) = \begin{cases} 0 & |x|<1 \\ \sum_{n=-\infty}^\infty \frac{\pi}{\ln(x)} \csc(\pi \bar{z}) a_{\bar{z}} & |x|>1 \end{cases}$$
Where $\bar{z} = \frac{2 \pi i (n+\frac{1}{2})}{\ln(x)}$.
Now, consider the particular case of $f_1(x) = \sum_{n=0}^\infty (\frac{1}{e})^{x^n}$. Let $f_2(x) = \sum_{n=0}^\infty \frac{(-1)^k}{k!}\frac{1}{x^k+1}$. Then we have have $f_1(x) = f_2(x)$ for $|x|<1$ (note that we need to evaluate $f_1(x)$ with cesaro summation). Both series still converge for real values of $x>1$. In this case, the difference between them is on the nose equal to $$\sum_{n=-\infty}^\infty \frac{\pi}{\ln(x)} \csc(\pi \bar{z}) a_{\bar{z}}$$ Where $a_k = \frac{1}{k!}$. In order to evaluate $a_k$ at complex values, we need to take the natural extension $a_k = \frac{1}{\Gamma(1+k)}$. If we change variables and remove some constant terms this sum becomes equal to $$\sum_{n=-\infty}^\infty \frac{\csc\left(\pi z(n+\frac{1}{2})\right)}{\Gamma\left(1+z(n+\frac{1}{2})\right)}$$ Which is the original sum being asked about. The goal would be to analytically continue this sum to get an idea of what $f_1(x)$ looks like on the domain $|x|>1$.