On sums of figurate numbers by using techniques of poset representation theory

TL;DR

This paper employs poset representation theory to analyze sums of figurate numbers, providing new criteria for specific sum representations and deriving identities related to Rogers-Ramanujan type formulas.

Contribution

It introduces a novel application of poset representation and differentiation algorithms to solve problems on figurate numbers and derive new identities.

Findings

Criteria for numbers as sums of three octahedral or polygonal numbers

Conditions for sums of four cubes with two equal

New Rogers-Ramanujan type identities involving figurate numbers

Abstract

We use representations and differentiation algorithms of posets, in order to obtain results concerning unsolved problems on figurate numbers. In particular, we present criteria for natural numbers which are the sum of three octahedral numbers, three polygonal numbers of positive rank or four cubes with two of them equal. Some identities of the Rogers-Ramanujan type involving this class of numbers are also obtained.