Bol's identity for skew-holomorphic Jacobi forms

Abstract

In this paper, we study an analogy of the heat operator to the skew-holomorphic Jacobi form case. Using this, we prove Bol's identity for skew-holomorphic Jacobi forms on $H_n\times M_{j,n}\left(C\right)$. This induces a map from skew-holomorphic Jacobi forms of weight $-k+\frac{n+j-1}{2}$ to those of weight $k+\frac{n+j-1}{2}+2$. When $n=j=1$, this map extends to skew-holomorphic harmonic Maass-Jacobi forms. In this case, we prove Zagier-type duality between Fourier coefficients of harmonic Maass-Jacobi forms and Fourier coefficients of weakly skew-holomorphic Jacobi forms.