The authors develop harmonic analysis associated to the family of operators \[ \Delta_{\vartheta}=\frac{1-|x|^2}{4}\left((1-|x|^2)\sum_1^n\partial_{x_j}^2-2\vartheta\sum_1^nx-j\partial_{x_j}+\vartheta(2-n-\theta)\right) \]
in a way parallel to the theory on real hyperbolic space. The generalized Helgason-Fourier transform and the \(\vartheta\)-spherical transform are studied. The authors prove an inversion formula and a Plancherel theorem for these transforms. Moreover, a formula for the heat kernel for \(\Delta_\vartheta\) is given.