On Rankin–Cohen brackets of Jacobi forms and Jacobi Poincaré series of matrix index

We prove that if a Rankin–Cohen bracket of two complex valued functions f and g on \({\mathcal {H}} \times \mathbb C^{(g\times 1)}\) $\mathcal{H}\times {\mathbb{C}}^{(g\times 1)}$ is a Jacobi form and if f is a Jacobi form and g has certain type of Fourier expansion, then g is also a Jacobi form. We also prove a relation between Rankin–Cohen brackets of certain Jacobi forms and Jacobi Poincaré series.