Given a region $\Omega \in \mathbb{R}^d$, consider the Poisson problem for the fractional Laplacian $(-\Delta)^{2s}$ with $s>0$. Given a function $g: \Omega^C \to \mathbb{R}$, what is the function $u: \mathbb{R}^d \to \mathbb{R}$ such that
$$\begin{cases} u =g & \in \Omega^C \\ (-\Delta)^{2s}u = 0 & \in \Omega \end{cases}$$
The restriction of $u$ to $\Omega$ can be found via integrating the Poisson kernel $P_{\Omega,s}:\Omega^C\times \Omega\to \mathbb{R}$. $$u(x) = \int_{\Omega^{C}}P_{\Omega,s}(y,x)g(y)dy$$
I was investigating the relationship between $P_{\Omega}$ and $P_{\Omega^C}$.
In particular, I noticed that if $\Omega$ was a sphere or a half space that $P_{\Omega^C,s}$ is the analytic continuation of $P_{\Omega,s}$ (i.e. they have the same functional form with different domains). For exmaple: if $B$ is the unit ball, $P_{B}$ and $P_{B^C}$ both have the form
$$\frac{\Gamma(\frac{d}{2})\sin(\pi s)}{\pi^{\frac{d}{2}+1}} (\frac{1-|y|^2}{|x|^2-1})^s\frac{1}{|x-y|^d}$$
Is this always true or simply a quirk of these particularly symmetric cases?
Any other references about the relationship between the Poisson kernels of complementary sets would be appreciated.