Pascal pyramid in the space ${\mathbf{H}}^2\text{\hspace{0.17em}⁣}\times \text{\hspace{0.17em}⁣}\mathbf{R}$

In this article we introduce a new type of Pascal pyramids. A regular squared mosaic in the hyperbolic plane yields a $(h^{2}r)$-cube mosaic in space ${\mathbf{H}}^2\text{\hspace{0.17em}⁣}\times \text{\hspace{0.17em}⁣}\mathbf{R}$ and the definition of the pyramid is based on this regular mosaic. The levels of the pyramid inherit some properties from the Euclidean and hyperbolic Pascal triangles. We give the growing method from level to level and show some illustrating figures.