Euler's partition theorem for all moduli and new companions to Rogers-Ramanujan-Andrews-Gordon identities

TL;DR

This paper generalizes Euler's partition theorem for all moduli, proves it for two families of partitions, and introduces new companions to classical Rogers-Ramanujan identities, supported by q-difference equations.

Contribution

It proposes a broad conjecture extending Euler's theorem, proves it for specific cases, and offers new identities related to Rogers-Ramanujan and Andrews-Gordon identities.

Findings

Conjecture generalizing Euler's theorem for all moduli.

Proof of the conjecture for two partition families.

Derivation of q-difference equations for the case of three moduli.

Abstract

In this paper, we give a conjecture, which generalises Euler's partition theorem involving odd parts and different parts for all moduli. We prove this conjecture for two family partitions. We give $q$-difference equations for the related generating function if the moduli is three. We provide new companions to Rogers-Ramanujan-Andrews-Gordon identities under this conjecture.