Looking for a new member of Gordon's identities

TL;DR

This paper offers a new algebraic perspective on Gordon's identities, extends them through conjectured families, and proves the conjecture for specific cases, advancing understanding of partition identities.

Contribution

It introduces a commutative algebra approach to Gordon's identities and proposes a conjectural extension, proven for cases r=2 and r=3.

Findings

Proved the conjecture for r=2 and r=3.

Established a new algebraic framework for Gordon's identities.

Suggested a broader family of partition identities.

Abstract

We give a commutative algebra viewpoint on Andrews recursive formula for the partitions appearing in "Gordon's identities", which are a generalization of Rogers-Ramanujan identities. Using this approach and differential ideals we conjecture a family of partition identities which extend Gordon's identities. This family is indexed by r >=2. We prove the conjecture for r=2 and r=3.