Likewise theta functions of rank $r$ on ${\mathbb{R}}^d$: analytic properties and associated Segal-Bargmann transform

We introduce and study the Hilbert space of $(L^2,\mathrm{\Gamma },\chi )$-likewise theta functions on ${\mathbb{R}}^d$ with respect to a given discrete subgroup $\mathrm{\Gamma }$ of arbitrary rank and a character $\chi$ of $\mathrm{\Gamma }$. A concrete description is given and an orthonormal basis is then constructed. Its range by the classical Segal-Bargmann transform is also characterized and leads to the so-called theta-Bargmann Fock space.