We introduce, in this paper, the harmonic analysis associated with the Heckman-Opdam-Jacobi operator on ${\mathbb{R}}^{d+1}$ in the W -invariant case. Next, we study the generalized wavelets and the generalized wavelet transform associated with this operator and we establish their properties. In particular, we prove for the generalized wavelet transform Plancherel and inversion formulas.