A proof of $J$-generalization of the Rogers-Ramanujan-Gordon identities via commutative algebra
Alapan Ghosh, Rupam Barman
TL;DR
The paper addresses proving a $J$-generalization of the Rogers-Ramanujan-Gordon identities through a commutative algebra approach. It extends Afsharijoo's $J=0$ proof to all $J\ge 0$ by linking the generating functions to Hilbert-Poincaré series of suitably constructed graded algebras. The method constructs a graded algebra $S=\mathbb{F}[x_1,x_2,\dots]$ and homogeneous ideals $P_{r,i,J}$ (along with related quotients) and derives recursion relations for their Hilbert-Poincaré series, following Coulson-type recursions. A $q$-adic limiting argument then yields $\mathcal{A}_{(r-1)J+\ell}=\mathcal{B}_{r,i,J}(q)$, thereby establishing the generalized identity and encompassing the classical $J=0$ case as the Rogers-Ramanujan-Gordon identities.
Abstract
The Rogers-Ramanujan-Gordon identities generalize the classical partition identities discovered independently by L. J. Rogers and S. Ramanujan. Recently, Afsharijoo gave a commutative algebra proof of the Rogers-Ramanujan-Gordon identities. In this article, we present a commutative algebra proof of a broader family of identities introduced by Coulson \textit{et al.}, which includes the Rogers-Ramanujan-Gordon identities as a special case. In the proof, we relate the generating functions associated with these identities to the Hilbert-Poincaré series of suitably constructed graded algebras.