  Let $\Gamma$ be a co-compact Fuchsian group of isometries on the Poincar\'e
disk $\DD$ and $\Delta$ the corresponding hyperbolic Laplace operator. Any
smooth eigenfunction $f$ of $\Delta$, equivariant by $\Gamma$ with real
eigenvalue $\lambda=-s(1-s)$, where $s={1/2}+ it$, admits an integral
representation by a distribution $\dd_{f,s}$ (the Helgason distribution) which
is equivariant by $\Gamma$ and supported at infinity $\partial\DD=\SS^1$. The
geodesic flow on the compact surface $\DD/\Gamma$ is conjugate to a suspension
over a natural extension of a piecewise analytic map $T:\SS^1\to\SS^1$, the
so-called Bowen-Series transformation. Let $\ll_s$ be the complex Ruelle
transfer operator associated to the jacobian $-s\ln |T'|$. M. Pollicott showed
that $\dd_{f,s}$ is an eigenfunction of the dual operator $\ll_s^*$ for the
eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic
eigenfunction $\psi_{f,s}$ of $\ll_s$ for the eigenvalue 1, given by an
integral formula \[ \psi_{f,s} (\xi)=\int \frac{J(\xi,\eta)}{|\xi-\eta|^{2s}}
  \dd_{f,s} (d\eta), \] \noindent where $J(\xi,\eta)$ is a $\{0,1\}$-valued
piecewise constant function whose definition depends upon the geometry of the
Dirichlet fundamental domain representing the surface $\DD/\Gamma$.
