  A series expansion for Heckman-Opdam hypergeometric functions
$\varphi_\lambda$ is obtained for all $\lambda \in \mathfrak a^*_{\mathbb C}.$
As a consequence, estimates for $\varphi_\lambda$ away from the walls of a Weyl
chamber are established. We also characterize the bounded hypergeometric
functions and thus prove an analogue of the celebrated theorem of Helgason and
Johnson on the bounded spherical functions on a Riemannian symmetric space of
the noncompact type. The $L^p$-theory for the hypergeometric Fourier transform
is developed for $0<p<2$. In particular, an inversion formula is proved when
$1\leq p <2$.
