On integral kernels for Dirichlet series associated to Jacobi forms (Q2874649)

Every Jacobi cusp form of weight \(k\) and index \(m\) over the semidirect product of \(\mathrm{SL}_2(\mathbb Z)\) and \( \mathbb Z^2\) corresponds to \(2m\) Dirichlet series constructed with its Fourier coefficients. This connection is generally described by a variation of the Mellin transform. The author describes a set of integral kernels which yield the \(2m\) Dirichlet series via the Petersson inner-product. It is shown that those kernels are Jacobi cusp forms and they are expressed in terms of Jacobi Poincaré series. Finally, a new proof is given of the analytic continuation and functional equations satisfied by these Dirichlet series.