Petersson norms of Jacobi–Eisenstein series and Gross–Kohnen–Zagier's formula

A regularized Petersson inner product on the space of Jacobi forms is defined and the regularized Petersson norms of Jacobi–Eisenstein series are computed. We use this result to establish Gross–Kohnen–Zagier's formula for Eisenstein series. In addition, we give an answer to the question raised by Böcherer and Das asking whether the regularized norm of Jacobi–Eisenstein series defined by them is non-zero. In the Supporting Information, we compute the Fourier coefficients of a suitable “new” basis of the space of Jacobi–Eisenstein series and give a remark on the proportional constant of the inner product formula in the theory of Jacobi forms.