Fourier-Bessel expansion (awaiting review)

Let $F$ be a Bianchi modular form of weight $2$ with respect to the congruence subgroup $\Gamma$ of a Bianchi group. As a function on hyperbolic 3-space $\mathcal{H}_3 = \{ (x,y) \in \C \times \R \mid y>0 \}$, $F$ is a periodic function in the variable $x\in\C$, and has a Fourier-Bessel expansion of the form $$F(x,y)=\sum_{0 \not =\alpha \in \mathcal{O}_K}c(\alpha) y^2 \mathbb{K}\left ( \dfrac{4\pi|\alpha|y}{\sqrt{|\triangle|}} \right ) \psi\left (\dfrac{\alpha x}{\sqrt{\triangle}} \right ),$$ where $$\psi(x)=e^{2\pi(x+\bar{x})}$$ and
$$\mathbb{K}(t)=\left ( -\dfrac{i}{2}K_1(y),K_0(y),\dfrac{i}{2}K_1(y) \right)$$ with $K_0,K_1$ are the hyperbolic K-Bessel functions satisfying the differential equation $$\dfrac{dK_j}{dy^2}+\dfrac{1}{y}\dfrac{dK_j}{dy}-\left ( 1+\dfrac{1}{y^{2j}}\right )K_j = 0, \ \ \ \ j=0,1$$ and decreasing rapidly at infinity.