About Pascal’s tetrahedron with hypercomplex entries

Abstract

It is evident, that the properties of monogenic polynomials in $(n+1)-$real variables significantly depend on the generators $e_1, e_2, \dots, e_n$ of the underlying $2^n$-dimensional Clifford algebra $Cl_{0,n}$ over $\mathbb{R}$ and their interactions under multiplication. The case of $n=3$ is studied through the consideration of Pascal's tetrahedron with hypercomplex entries as special case of the general Pascal simplex for arbitrary $n$, which represents a useful geometric arrangement of all possible products. The different layers ${\mathcal{L}_k$ of Pascal's tetrahedron (or pyramid) are built by ordered symmetric products contained in the trinomial expansion of $(e_1+e_2+e_3)^k$, $k=0,1,\dots$.