Marking and shifting a part in partition theorems

TL;DR

This paper introduces refined analytic and combinatorial versions of classical partition theorems, incorporating marking and shifting of parts to generalize identities like Rogers-Ramanujan and Euler's theorem.

Contribution

It provides novel generalizations of key partition theorems by allowing parts to be marked or shifted, expanding the combinatorial framework.

Findings

Refined identities for Rogers-Ramanujan, Gollnitz-Gordon, Euler's, and Andrews-Gordon theorems.

Generalizations involving marking a single part or a sum of parts.

Analytic and combinatorial proofs of the generalized identities.

Abstract

Refined versions, analytic and combinatorial, are given for classical integer partition theorems. The examples include the Rogers-Ramanujan identities, the Gollnitz-Gordon identities, Euler's odd=distinct theorem, and the Andrews-Gordon identities. Generalizations of each of these theorems are given where a single part is "marked" or weighted. This allows a single part to be replaced by a new larger part, "shifting" a part, and analogous combinatorial results are given in each case. Versions are also given for marking a sum of parts.