  For odd dimensional Poincar\'e-Einstein manifolds $(X^{n+1},g)$, we study the
set of harmonic $k$-forms (for $k<\ndemi$) which are $C^m$ (with $m\in\nn$) on
the conformal compactification $\bar{X}$ of $X$. This is infinite dimensional
for small $m$ but it becomes finite dimensional if $m$ is large enough, and in
one-to-one correspondence with the direct sum of the relative cohomology
$H^k(\bar{X},\pl\bar{X})$ and the kernel of the Branson-Gover \cite{BG}
differential operators $(L_k,G_k)$ on the conformal infinity
$(\pl\bar{X},[h_0])$. In a second time we relate the set of
$C^{n-2k+1}(\Lambda^k(\bar{X}))$ forms in the kernel of $d+\delta_g$ to the
conformal harmonics on the boundary in the sense of \cite{BG}, providing some
sort of long exact sequence adapted to this setting. This study also provides
another construction of Branson-Gover differential operators, including a
parallel construction of the generalization of $Q$ curvature for forms.
