  The aim of this paper is to give the characterisation of the L p Range (p
$\ge$ 2) of the Generalised Poisson Transform of the Hyperbolic space B(H n),
(n $\ge$ 2), over the classical field of the quaternions H. Namely, if f is an
hyperfunction in the boundary of B(H n), then we show that f is in L p
($\partial$B(H n)) if and only if it's generalised poisson transform satisfy an
Hardy type growth condition. An explicit expression of the generalized
spherical functions is given. Mathematics Subject Classification (2010).
Primary 22E46; Secondary 33Cxx.
