On Taylor Expansion of Jacobi Forms of Half-Integral Weight

In this paper we generalize the Taylor development map for Jacobi forms of half-integral weight and obtain its adjoint map with respect to the Petersson scalar product. We observe that the study of the Taylor development in the case of half-integral weight Jacobi forms gives several interesting identities. Finally, we show how the Taylor development map occurs in a connection between theta series of degree 2 and degree 1. As a consequence, we prove that Maass space of degree 2 contains theta series if and only if k = 4 ork = 8.