Parity of the coefficients of certain eta-quotients, III: two special classes

TL;DR

This paper investigates the parity properties of specific eta-quotients, including those related to Andrews' singular overpartitions and pure eta-powers, contributing to understanding their implications for partition functions.

Contribution

It advances the study of eta-quotients by analyzing two special classes, expanding knowledge on their parity and density results.

Findings

Analyzed eta-quotients of the form f_t^3/f_1 and their parity properties.

Investigated parity of pure eta-powers f_1^t and added new density theorems.

Provided implications for the parity of the partition function and related q-series.

Abstract

We continue a series of papers studying the parity of families of eta-quotients, which provide implications for the parity of the partition function as well as an overarching conjecture on related $q$-series. The present article focuses on two classes. One consists of eta-quotients of the form $f_t^3/f_1$, a distinguished case of Andrews' singular overpartitions that has recently attracted attention among researchers. In addition, we investigate the parity of certain pure eta-powers $f_1^t$, appending new results to known density theorems.