If \(f\) and \(g\) are two modular forms of weight \(k\) and \(\ell \), then their \(n\)-th Rankin-Cohen bracket \[ [f,g]_n=\sum _{r+s=n}[-1)^r \binom{k+n-1}{s}\binom{\ell +n-1}{r} \frac{d^rf}{dz^r}\frac{d^sg}{dz^s} \] is a modular form of weight \(k+\ell +2n\). These Rankin-Cohen brackets afford an explicit decomposition of the tensor product of two representations of the group \(SL(2,\mathbb{R})\) belonging to the holomorphic discrete series. The author of the present paper generalizes these results by considering the setting of a Hermitian symmetric space of tube-type. In fact the simplest one is the upper complex half-plane. In this setting generalized Rankin-Cohen brackets have been defined these last years. The author establishes new formulae for them. For a symmetric cone \(\Omega \), and the associated tube \(T_{\Omega }\), one defines a family \(\mathcal{H}_{\lambda }(T_{\Omega })\) of Hilbert spaces of holomorphic functions on the tube \(T_{\Omega }\). The group \(G(T_{\Omega })\) of holomorphic automorphisms of \(T_{\Omega }\), or more precisely the covering group \(\tilde G\) of its identity component, acts on \(\mathcal{H}_{\lambda }(T_{\Omega })\) as an irreducible unitary representation \(\pi _{\lambda }\) which belongs to the holomorphic discrete series of scalar type. The generalized Rankin-Cohen bracket \(B_{\lambda ,\mu }^{(k)}\) is a bidifferential operator which maps \(\mathcal{H}_{\lambda }(T_{\Omega })\otimes \mathcal{H}_{\mu }(T_{\Omega })\) into \(\mathcal{H}_{\lambda +\mu +2k}(T_{\Omega })\), with a covariance property with respect to \(\tilde G\). For the proofs the author considers the reproducing kernels of the spaces \(\mathcal{H}_{\lambda }(T_{\Omega })\), and uses also the \(L^2\)-model for the representation \(\pi _{\lambda }\). As a biproduct of these proofs, multivariate Jacobi polynomials show up with a generalized Rodrigues formula.