Frame Sets and Zeros of Zak Transforms of Extended Gaussians

Let $a,b,c\in\mathbb C$ with $\re(a)<0$, we show that the extended Gaussian $e^{ax^2+bx+c}$ has maximal frame set (i.e., its frame set consists of precisely all positive pairs $(α,β)$ with $αβ<1$), and its Zak transform has a unique simple zero in the unit square $[0,1)^2$ (in particular, the zero i