${\mathbb{H}}=\{\tau\in {\mathbb{C}};\;\Im(\tau)>0\}$ by linear fractional transformations. We write $\Gamma_\infty'=\{{\left(\begin{smallmatrix} 1 & n \\ 0 & 1\end{smallmatrix}\right) };\;n\in {\mathbb{Z}}\}$. As usual we abbreviate $e(x)=e^{2\pi i x}$. For a positive integer $m$ we define the Poincaré series of weight $0$ and index $m$ by $$\label{DefF}
F_{m}(\tau,s)=\sum_{\gamma\in \Gamma'_\infty{\backslash}\Gamma'} {\mathcal{I}}_s\big(2\pi m \Im(\gamma \tau)\big) e\big(-m \Re(\gamma \tau)\big),$$ where $\tau=x+iy\in{\mathbb{H}}$ and $s\in{\mathbb{C}}$ with $\Re(s)>1$. It converges normally for $\Re(s)>1$ and defines a $\Gamma'$-invariant function on ${\mathbb{H}}$. It is an eigenfunction of the hyperbolic Laplacian with eigenvalue $s(s-1)$.

\[fourierF\] The Poincaré series $F_{m}(\tau,s)$ has the Fourier expansion $$\begin{aligned}
F_{m}(\tau,s)&=\left(2{\mathcal{I}}_{s}(2\pi m y)+{\mathcal{K}}_s(2\pi m y)\right) e(-mx)\\
&\phantom{=}{}+ b_m(0,s) y^{1-s}  + \sum_{\substack{ n\in {\mathbb{Z}}\setminus \{0\}}} b_m(n,s) {\mathcal{K}}_s(2\pi ny) e(nx),\end{aligned}$$ where $$b_m(n,s)=\begin{cases}
\displaystyle 2\pi \left| \frac{m}{n}\right|^{1/2} \sum_{c=1}^\infty  H_c(m,n) I_{2s-1}\!\left(\frac{4\pi}{c}\sqrt{|mn|}\right), & \text{$n>0$,}\\[3ex]
\displaystyle \frac{4 \pi^{1+s} m^{s}}{(2s-1)\Gamma(s)} \sum_{c=1}^\infty c^{1-2s} H_c(m,0), & \text{$n=0$,}\\[3ex]
\displaystyle -\delta_{-m,n} + 2\pi \left| \frac{m}{n}\right|^{1/2} \sum_{c=1}^\infty H_c(m,n)
J_{2s-1}\!\left(\frac{4\pi}{c}\sqrt{|mn|}\right), & \text{$n<0$}.
\end{cases}$$ Here $ H_c(m,n)$ denotes the Kloosterman sum $$\label{DefH_c}
H_c(m,n)=\frac{1}{c} \sum_{\substack{d(c)^*}} e\!\left( \frac{nd-m\bar d}{c}\right),$$ where the sum runs through the multiplicative group $({\mathbb{Z}}/c{\mathbb{Z}})^*$ and $\bar d$ denotes the multiplicative inverse of $d$. Moreover, $J_\nu(z)$ and $I_\nu(z)$ are the usual Bessel functions as defined in [@AS] §9.

This is a special case of [@Br2] Theorem 1.9. See also [@Ni] or [@He].

