Let $D$ be the definite quaternion division algebra over $\mathbb{Q}$. $\mathcal{O}$ is a maximal order inside $D$, let's fix $\mathcal{O}$ to be the Hurwitz quaternion. Let $\Gamma=PGL_2(\mathcal{O})$ and $\Gamma_\infty$ is the stabilizer of the unique cusp of $\Gamma\backslash PGL_2(\mathbb{H})/K$, where $\mathbb{H}=D\otimes_\mathbb{Q}\mathbb{R}$ the Hamilton's quaternions.

I want to find a literature regarding analytic continuation and functional equation for the Eisenstein series related to $\Gamma_\infty\backslash\Gamma$. Intuitively it may look like a mimic proof of the cases $SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})$ or $SL_2(\mathbb{Z[i]})\backslash SL_2(\mathbb{C})$, but a significant problem is arising regarding non-commutativity of $\mathcal{O}$.

One could possibly get away using Langlands's work, but we are still in "rank $1$ and unique cusp" case I am expecting an easier way than Langlands, something like Colin de Verdière's work for $SL_2(\mathbb{R})$.

Thanks for any help and reference.