Certain positive $q$-series and inequalities for two-color partitions

TL;DR

This paper investigates specific $q$-series related to two-color partitions, exploring their positivity and deriving inequalities based on their generating functions, with some cases confirmed and others remaining uncertain.

Contribution

It introduces new $q$-series linked to two-color partitions and analyzes their positivity, providing inequalities and highlighting unresolved cases for certain parameters.

Findings

Positivity holds for initial values of (k,m)

Generated series count weighted two-color partitions

Inequalities are derived from series positivity

Abstract

We consider some $q$-series which depend on a pair of positive integers $(k,m)$. While positivity of these series holds for the first few values of $(k,m)$, the situation is quite unclear for other values of $(k,m)$. In addition, our series generate the number of certain two-color integer partitions weighted by $(-1)^j$ where $j$ is the number of even parts. Therefore, inequalities involving these partitions will be deduced from the positivity of their generating functions.