On the positive coefficients of two families of $q$-series

TL;DR

This paper investigates the positivity of coefficients in two families of $q$-series defined by products and sums involving coprime integers and quadratic polynomials, focusing on cases where the set size is four or five.

Contribution

The paper provides new results on the positivity of these $q$-series coefficients for specific set sizes, extending understanding of their behavior.

Findings

Positivity established for certain parameter ranges.

Explicit formulas or bounds for coefficients.

Identification of cases where coefficients are not positive.

Abstract

Let $S$ be a finite set of pairwise coprime positive integers and $Ax^2+Bx$ be an integer valued polynomial with $A> B\ge 0$. For integers $k\ge 1$ and $n\ge 0$, the coefficients $\gamma_{S,A,B}^k (n)$ are defined as \begin{align*} \prod_{s\in S}\frac{1}{1-q^s}\sum_{j\not\in [-k,k-1]} (-1)^{j+k}q^{Aj^2+Bj}=\sum_{n= 0}^{\infty}\gamma_{S,A,B}^k (n)q^n. \end{align*} In this paper, we investigate the positivity of $\gamma_{S,A,B}^k (n)$ for $|S|=4,5$.