Asymptotic formulae for Eulerian series

Let $(a;q{)}_{\mathrm{\infty }}$ be the $q$-Pochhammer symbol and ${\mathrm{l}\mathrm{i}}_2(x)$ be the dilogarithm function. Let ${\prod }_{\alpha ,\beta ,\gamma }$ be a finite product with every triple $(\alpha ,\beta ,\gamma )\in ({\mathbb{R}}_{>0}{)}^3$ and $S_{\alpha \beta \gamma }\in \mathbb{R}$. Also let the triple $(A,B,v)\in ({\mathbb{R}}_{>0}\times {\mathbb{R}}^2)\cup (\{0{\}}^2\times {\mathbb{R}}_{>0})\cup (\{0\}\times {\mathbb{R}}_{<0}\times \mathbb{R})$. In this work, we let $z=e^v$, denote by $H_{-1}(u)=vu-Au^2+\sum_{\alpha}$\mathrm{l}\mathrm{i}$2(e^{-\alpha u})\sum{\beta,\gamma} \beta^{-1}S_{\alpha\beta\gamma}$ and consider the Eulerien series $$\mathcal{H}(z;q)=\sum _{m=0}^{\mathrm{\infty }}\frac{q^{A{m}^2+Bm}{z}^m}{\prod _{\alpha ,\beta ,\gamma }(q^{\alpha m+\gamma };q^{\beta }{)}_{\mathrm{\infty }}^{S_{\alpha \beta \gamma }}}\mathrm{.}$$ We prove that if there exist an $\epsilon >0$ such that $H_{-1}(u)$ is an increasing function on $[0,\epsilon )$, then as $q\to 1^{-}$, $$\mathcal{H}(z;q)=(1+o(\mathrm{\mid }\log q{\mathrm{\mid }}^p))\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{q^{A{x}^2+Bx}{z}^x}{\prod _{\alpha ,\beta ,\gamma }(q^{\alpha x+\gamma };q^{\beta }{)}_{\mathrm{\infty }}^{S_{\alpha \beta \gamma }}}dx$$ holds for each $p\ge 0$. We also obtain full asymptotic expansions for $\mathcal{H}(z;q)$ which satisfy above condition as $q\to 1^{-}$. The complete asymptotic expansions for related basic hypergeometric series could be derived as special cases.