Rankin–Cohen brackets on Jacobi forms and the adjoint of some linear maps

Abstract

Given a fixed Jacobi cusp form, we consider a family of linear maps between the spaces of Jacobi cusp forms using the Rankin–Cohen brackets, and then we compute the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Jacobi cusp forms constructed using this method involve special values of certain Dirichlet series associated to Jacobi cusp forms. This is a generalization of the work due to Kohnen (Math Z, 207:657–660, 1991) and Herrero (Ramanujan J, 10.1007/s11139-013-9536-5, 2014) in case of elliptic modular forms to the case of Jacobi cusp forms which is also considered earlier by Sakata (Proc Japan Acad Ser A, Math Sci 74, 1998) for a special case.