Alapan Ghosh | Searcharxiv

arXiv2026

$J$-generalization of the Rogers-Ramanujan-Gordon identities via commutative algebra

The Rogers-Ramanujan-Gordon identities generalize the classical partition identities discovered independently by L. J. Rogers and S. Ramanujan. In 2021, Afsharijoo provided a commutative algebra proof of the Rogers-Ramanujan-Gordon identities. Building on the Afsharijoo's approach, 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-Poincar\'e series of suitably constructed graded algebras.

math.CO↗

arXiv2026

On the G\"ollnitz-Gordon-Andrews identities via commutative algebra

The G\"ollnitz-Gordon-Andrews identities generalize the classical partition identities discovered independently by H. G\"ollnitz and B. Gordon. These are Rogers-Ramanujan-type identities involving generating functions of partitions satisfying certain kinds of difference conditions on the one hand and infinite periodic products on the other. In 2021, Afsharijoo provided a commutative algebra proof of the Rogers-Ramanujan-Gordon identities. Building on Afsharijoo's approach, we investigate the G\"ollnitz-Gordon-Andrews identities using techniques from commutative algebra. More generally, we establish a broader family of identities, of which the G\"ollnitz-Gordon-Andrews identities arise as special cases. Our approach interprets the associated generating functions in terms of Hilbert-Poincar\'e series of suitably constructed graded algebras, providing the first commutative algebra framework for these identities.

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