Double series representations for Schur's partition function and related identities

TL;DR

This paper introduces new double summation hypergeometric $q$-series representations for partition functions related to G"ollnitz, Gordon, and Schur identities, using multiple proof techniques.

Contribution

It provides novel double series formulas for partition functions and explores their connections through various proof methods, enriching the theory of $q$-series and partition identities.

Findings

New double summation hypergeometric $q$-series representations for partition functions.

Multiple proof techniques including bijections, modular diagrams, and $q$-difference equations.

Identification of interesting special cases within a general family of double series.

Abstract

We prove new double summation hypergeometric $q$-series representations for several families of partitions, including those that appear in the famous product identities of G\"ollnitz, Gordon, and Schur. We give several different proofs for our results, using bijective partitions mappings and modular diagrams, the theory of $q$-difference equations and recurrences, and the theories of summation and transformation for $q$-series. We also consider a general family of similar double series and highlight a number of other interesting special cases.