A series expansion for Heckman-Opdam hypergeometric functions ϕ_λ is obtained for all λ ∈ a*_C . As a consequence, estimates for ϕ_λ 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 ≤ p < 2.