The Sylvester Theorem and the Rogers-Ramanujan Identities over Totally Real Number Fields

TL;DR

This paper generalizes classical partition identities, including Sylvester's Theorem and Rogers-Ramanujan identities, to the setting of totally real number fields, providing new algebraic and combinatorial insights.

Contribution

It extends key partition identities to totally real algebraic integers, offering new versions and generalizations in algebraic number theory.

Findings

Proved generalized Sylvester Theorem over totally real fields

Established new versions of Rogers-Ramanujan identities

Connected classical identities with algebraic integers in number fields

Abstract

In this paper, we prove two identities on the partition of a totally positive algebraic integer over a totally real number field which are the generalization of the Sylvester Theorem and that of the Rogers-Ramanujan Identities. Additionally, we give an another version of generalized Rogers-Ramanujan Identities.