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

Let $S$ be a finite set of pairwise coprime positive integers and $A{x}^2+Bx$ 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}γ_{S,A,B}^k (n)q^n. \end{align*} In this paper, we investigate the positivity of ${\gamma }_{S,A,B}^k(n)$ for $\mathrm{\mid }S\mathrm{\mid }=4,5$.