y_2 mp }\right),\end{aligned}$$ where $Q_{s-1}(t)$ is the Legendre function of the second kind defined by (cf. [@AS] §8) $$Q_{s-1}(t) =\int_0^\infty (t +\sqrt{t^2-1} \cosh v)^{-s} dv\qquad (t>1, \,\Re(s)>0).$$

The sum converges normally for $(z_1,z_2)\in  {\mathbb{H}}^2\setminus \tilde T_m$ and $s\in {\mathbb{C}}$ with $\Re(s)>1$. We will continue it to a neighborhood of $s=1$ by computing the Fourier expansion. To this end we write $$\begin{aligned}
\label{auft}
\tilde\Phi_{mp^2}(z_1,z_2,s)&= \tilde\Phi^0_{mp^2}(z_1,z_2,s) +
2\sum_{a=1}^{\infty} \tilde\Phi^a_{mp^2}(z_1,z_2,s)\\ \intertext{with}
\tilde\Phi^a_{mp^2}(z_1,z_2,s)&= \sum_{\substack{ b\in{\mathbb{Z}}\\
\lambda\in\partial_F^{-1}\\ ab-{\operatorname{N}}(\lambda)=mp }}
\psi{\left(\begin{smallmatrix} a & \lambda \\ \lambda' & b\end{smallmatrix}\right) } Q_{s-1}\left(1+\frac{ |az_1 z_2
+\lambda z_1+\lambda' z_2+b|^2}{ 2y_1 y_2 mp }\right). \label{Phia}\end{aligned}$$ Note that the partial sums $\tilde\Phi^a_{mp^2}(z_1,z_2,s)$ converge normally for $\Re(s)>1/2$.

For $a=0$ we have $$\tilde\Phi^0_{mp^2}(z_1,z_2,s)= \sum_{\substack{ b\in{\mathbb{Z}}\\ \lambda\in\partial_F^{-1}\\ {\operatorname{N}}(\lambda)=-mp }}
\epsilon_p(b)  Q_{s-1}\left(1+\frac{ |\lambda z_1+\lambda' z_2+b|^2}{ 2y_1 y_2 mp }\right).$$ If $a$ is a positive integer coprime to $p$, then $$\begin{aligned}
\tilde\Phi^a_{mp^2}(z_1,z_2,s)&= \epsilon_p(a)  \sum_{\substack{ b\in{\mathbb{Z}}\\ \lambda\in\partial_F^{-1}\\ ab-{\operatorname{N}}(\lambda)=mp }} Q_{s-1}\left(1+\frac{ |az_1 z_2 +\lambda z_1+\lambda' z_2+b|^2}{ 2y_1 y_2 mp }\right)\\
&= \epsilon_p(a) \Phi^a_{mp^2}(z_1,z_2,s),\end{aligned}$$ where $\Phi^a_{mp^2}(z_1,z_2,s)$ is the function defined in [@Br1] (see §3 equation (16)). If $a$ is divisible by $p$, then $$\begin{aligned}
\tilde\Phi^a_{mp^2}(z_1,z_2,s)&= \sum_{\substack{ b\in{\mathbb{Z}}\\ \lambda\in\partial_F^{-1}\\ ab-{\operatorname{N}}(\lambda)=mp }}
\epsilon_p(b)  Q_{s-1}\left(1+\frac{ |az_1 z_2 +\lambda z_1+\lambda' z_2+b|^2}{ 2y_1 y_2 mp }\right).\end{aligned}$$

We compute the Fourier expansion of $\tilde\Phi^a_{mp^2}(z_1,z_2,s)$ in these 