A proof of the G\"ollnitz-Gordon-Andrews identities via commutative algebra

TL;DR

This paper provides a commutative algebra proof of the G"ollnitz-Gordon-Andrews identities, connecting their generating functions to Hilbert-Poincaré series of graded algebras, and establishes a broader family of related identities.

Contribution

It introduces a novel algebraic proof technique for these identities and generalizes them within a new family of related identities.

Findings

Established a family of identities generalizing G"ollnitz-Gordon-Andrews identities

Connected generating functions to Hilbert-Poincaré series of graded algebras

Provided a new algebraic proof approach for partition identities

Abstract

The G\"ollnitz-Gordon-Andrews identities generalize the partition identities discovered independently by H. G\"ollnitz and B. Gordon. In this article, we present a commutative algebra proof of the G\"ollnitz-Gordon-Andrews identities. More generally, we establish a family of identities, the special cases of which are the G\"ollnitz-Gordon-Andrews identities. In the proof, we relate the generating functions associated with these identities to the Hilbert-Poincar\'e series of suitably constructed graded algebras.