On three-dimensional expansions of the polynomial function f(n)=n^m

Abstract

In this article (points 3,4,5) we will study 3-D expansions of the polynomial function f(n)=n^m, for n,m natural numbers. These 3-D expansions are somewhat similar to the triangular (2-D) expansions of these functions, but here we consider here more index variables in the expansion, i.e. exactly 3 instead of 2. Additionally potential 3-dimensional expansions involve a different set of polynomial powers. In particular, only odd powers m >1 one can develop in a 2-D way. In the 3-D case we can only consider powers of the form m=3l+2 for l >0. The problem of the existence of triangular expansions of discrete polynomial functions was first raised by Petro Kolosov in February 2018. Triangular expansions are presented in general in the first two parts of the article. At the end (point 6), additional questions related to three dimensional expansions are presented.