  Let $(m_1,\ldots,m_J)$ and $(r_1,\ldots,r_J)$ be two sequences of $J$
positive integers satisfying $1\le r_j< m_j$ for all $j=1,\ldots,J$. Let
$(\delta_1,\ldots,\delta_J)$ be a sequence of $J$ nonzero integers. In this
paper, we study the asymptotic behavior of the Taylor coefficients of the
infinite product $$\prod_{j=1}^J\Bigg(\prod_{k\ge
1}\big(1-q^{r_j+m_j(k-1)}\big)\big(1-q^{-r_j+m_jk}\big)\Bigg)^{\delta_j}.$$
