  A generalized Fock-Bargmann-Hartogs domain $D_n^{\mathbf{m},\mathbf{p}}$ is
defined as a domain fibered over $\mathbb{C}^{n}$ with the fiber over $z\in
\mathbb{C}^{n}$ being a generalized complex ellipsoid
$\Sigma_z({\mathbf{m},\mathbf{p}})$. In general, a generalized
Fock-Bargmann-Hartogs domain is an unbounded non-hyperbolic domains without
smooth boundary. The main contribution of this paper is as follows. By using
the explicit formula of Bergman kernels of the generalized
Fock-Bargmann-Hartogs domains, we obtain the rigidity results of proper
holomorphic mappings between two equidimensional generalized
Fock-Bargmann-Hartogs domains. We therefore exhibit an example of unbounded
weakly pseudoconvex domains on which the rigidity results of proper holomorphic
mappings can be built.
