Asymptotics for the Taylor coefficients of certain infinite products

Let $(m_1,\dots ,m_J)$ and $(r_1,\dots ,r_J)$ be two sequences of $J$ positive integers satisfying $1\le r_j<m_j$ for all $j=1,\dots ,J$. Let $({\delta }_1,\dots ,{\delta }_J)$ be a sequence of $J$ nonzero integers. In this paper, we study the asymptotic behavior of the Taylor coefficients of the infinite product $$\prod _{j=1}^J\left(\prod _{k\ge 1}\right(1-q^{r_j+m_j(k-1)}\left)\right(1-q^{-r_j+m_{j}k}){)}^{{\delta }_j}\mathrm{.}$$