Infinitely many negative terms in two weighted partition sequences
Recently, Andrews and El Bachraoui introduced partition-counting functions $a_2(n)$ and $b_2(n)$, together with their signed analogues $a'_2(n)$ and $b'_2(n)$, and proved that $a'_2(n), b'_2(n)>0$ for all $n\geq 3$. For $m\geq 2$, the latter functions extend naturally to $a'_m(n)$ and $b'_m(n)$, defined by \begin{align*} \sum_{n\geq 0} a'_m(n)q^n &= -\frac{q}{1-q} \frac{(q^m;q^m)_\infty}{(q;q^m)_\infty} +\frac{q}{(1-q)^2},\\ \sum_{n\geq 0} b'_m(n)q^n &= -\frac{q(1-q^{m-1})}{1-q} \frac{(q^{2m};q^m)_\infty}{(q;q^m)_\infty} +\frac{q}{1-q} \left( \frac{1}{1-q} -\frac{q^{m-1}}{1-q^{m+1}} \right). \end{align*} They asked whether there exists an integer $m>2$ for which $a'_m(n)$ is eventually positive, and posed the same question for $b'_m(n)$. In this paper, we resolve these two questions.