For odd dimensional Poincar\'e-Einstein manifolds
, we study the set of harmonic
k-forms (for
k<\ndemi) which are
(with
m\in\nn) on the conformal compactification
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
... [Show full abstract] H^k(\bar{X},\pl\bar{X}) and the kernel of the Branson-Gover \cite{BG} differential operators on the conformal infinity (\pl\bar{X},[h_0]). In a second time we relate the set of forms in the kernel of 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.