On integral kernels for Dirichlet series associated to Jacobi forms

Every Jacobi cusp form of weight k and index m over SL2(Z) (sic) Z(2) is in correspondence with 2m Dirichlet series constructed with its Fourier coefficients. The standard way to get from one to the other is by a variation of the Mellin transform. In this paper, we introduce a set of integral kernels which yield the 2m Dirichlet series via the Petersson inner product. We show that those kernels are Jacobi cusp forms and express them in terms of Jacobi Poincar'e series. As an application, we give a new proof of the analytic continuation and functional equations satisfied by the Dirichlet series mentioned above.