Wei-Norman and Berezin's equations of motion on the Siegel-Jacobi disk

We show that the Wei-Norman method applied to describe the evolution on the Siegel-Jacobi disk $\mathcal{D}^J_1=\mathcal{D}_1\times\mathbb{C}^1$, where $\mathcal{D}_1$ denotes the Siegel disk, determined by a hermitian Hamiltonian linear in the generators of the Jacobi group $G^J_1$ and Berezin's scheme using coherent states give the same equations of quantum and classical motion when are expressed in the coordinates in which the K\"ahler two-form $\omega_{\mathcal{D}^J_1}$ can be written as $\omega_{\mathcal{D}^J_1}=\omega_{\mathcal{D}_1}+\omega_{\mathbb{C}^1}$. The Wei-Norman equations on $\mathcal{D}^J_1$ are a particular case of equations of motion on the Siegel-Jacobi ball $\mathcal{D}^J_n$ generated by a hermitian Hamiltonian linear in the generators of the Jacobi group $G^J_n$ obtained in Berezin's approach based on coherent states on $\mathcal{D}^J_n$.