A generalization of vector valued Jacobi forms

Abstract

The Fourier--Jacobi coefficients of vector valued Siegel modular forms of degree $n$ are more general functions than vector valued Jacobi forms defined by Ziegler [9] even when $n=2$. We define generalized vector valued Jacobi forms corresponding to the above coefficients when $n=2$ and prove that such a space is isomorphic to a certain product of spaces of usual scalar valued Jacobi forms of various weights. This isomorphism is realized by certain linear holomorphic differential operators. The half-integral weight case is also treated.