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