Non-vanishing of Jacobi Poincar\'{e} series

We prove that under suitable conditions, the Jacobi Poincar'{e} series of exponential type of integer weight and matrix index does not vanish identically. For classical Jacobi forms, we construct a basis consisting of the “first” few Poincar'{e} series and also give conditions both dependent and independent of the weight, which ensures non-vanishing of classical Jacobi Poincar'{e} series. Equality of certain Kloosterman-type sums is proved. Also, a result on the non-vanishing of Jacobi Poincar'{e} series is obtained when an odd prime divides the index.