Isomorphism between Jacobi forms of index $D_{2n+1}$ and elliptic modular forms of level $2$

This paper has three main objectives: (i) To establish an isomorphism between Jacobi forms of index $D_{2n+1}$ (lattice index) and elliptic modular forms of level $2$. (ii) To provide an explicit formula for the Fourier coefficients of Jacobi--Eisenstein series of index $D_{2n+1}$. (iii) To construct a holomorphic modular form of weight $3/2$ and level $8$ (and $4$) from the Zagier--Eisenstein series $\mathscr{F}$ of weight $3/2$ and level $4$. Moreover, we show that the four functions $E^*_2$, $\eta^3$, $\theta^3$ and $\mathscr{F}$ have essentially the same Hecke eigenvalue $1+p$ for any odd prime $p$, where $E^*_2$ is the non-holomorphic Eisenstein series of weight $2$, $\eta$ is the Dedekind eta-function and $\theta$ is the usual theta function. This fact arises as a special case of the isomorphism of (i).