The group SU(1,d) acts naturally on the Hilbert space $L²\left(Bd{\mu }_{\alpha }\right)(\alpha >-1)$, where B is the unit ball of ${ℂ}^d$ and $d{\mu }_{\alpha }$ the weighted measure ${(1-|z\left|²\right)}^{\alpha }dm\left(z\right)$. It is proved that the irreducible decomposition of the space has finitely many discrete parts and a continuous part. Each discrete part corresponds to a zero of the generalized Harish-Chandra c-function in the lower half plane. The discrete parts are studied via invariant Cauchy-Riemann operators. The representations on the discrete parts are equivalent to actions on some holomorphic...