  We derive an integral representation for the Jacobi-Poisson kernel valid for
all admissible type parameters $\alpha,\beta$ in the context of Jacobi
expansions. This enables us to develop a technique for proving standard
estimates in the Jacobi setting, which works for all possible $\alpha$ and
$\beta$. As a consequence, we can prove that several fundamental operators in
the harmonic analysis of Jacobi expansions are (vector-valued)
Calder\'on-Zygmund operators in the sense of the associated space of
homogeneous type, and hence their mapping properties follow from the general
theory. The new Jacobi-Poisson kernel representation also leads to sharp
estimates of this kernel. The paper generalizes methods and results existing in
the literature, but valid or justified only for a restricted range of $\alpha$
and $\beta$.
