Optimal ball and horoball packings generated by $3$-dimensional simply truncated Coxeter orthoschemes with parallel faces

In this paper we consider the ball and horoball packings belonging to $3$-dimensional Coxeter tilings that are derived by simply truncated orthoschemes with parallel faces. The goal of this paper to determine the optimal ball and horoball packing arrangements and their densities for all above Coxeter tilings in hyperbolic 3-space $\mathbb{H}^3$. The centers of horoballs are required to lie at ideal vertices of the polyhedral cells constituting the tiling, and we allow horoballs of different types at the various vertices. We prove that the densest packing of the above cases is realized by horoballs related to $\{\infty;3;6;\infty \}$ and $\{\infty;6;3;\infty \}$ tilings with density $\approx 0.8413392$.