Pairings of harmonic Maass-Jacobi forms involving special values of partial L-functions

We prove that, for a given Jacobi integral F, there is a harmonic Maass-Jacobi form such that its holomorphic part is F, and that the converse is also true. As an application, we construct a pairing between two Jacobi integrals that is defined by special values of partial L-functions of skew-holomorphic Jacobi cusp forms. We obtain connections between this pairing and the Petersson inner product for skew-holomorphic Jacobi cusp forms. This result can be considered as an analogue of the Haberland formula of elliptic modular forms for Jacobi forms.