arXiv2017
Let $(a;q)_{\infty}$ be the $q$-Pochhammer symbol and $\mathrm{li}_2(x)$ be the dilogarithm function. Let $\prod_{α,β,γ}$ be a finite product with every triple $(α,β,γ)\in(\mathbb{R}_{>0})^3$ and $S_{αβγ}\in\mathbb{R}$. 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 $H_{-1}(u)=vu-Au^2+\sum_α\mathrm{li}_2(e^{-αu})\sum_{β,γ} β^{-1}S_{αβγ}$ and consider the Eulerien series \[\mathcal{H}(z;q)=\sum_{m=0}^{\infty}\frac{q^{Am^2+Bm}z^{m}}{\prod\limits_{α,β,γ}(q^{αm+γ};q^β)_{\infty}^{S_{αβγ}}}.\] We prove that if there exist an $\varepsilon>0$ such that $H_{-1}(u)$ is an increasing function on $[0,\varepsilon)$, then as $q\rightarrow 1^-$, \[\mathcal{H}(z;q)=\left(1+o\left(|\log q|^p\right)\right)\int\limits_{0}^{\infty}\frac{q^{Ax^2+Bx}z^{x}}{\prod\limits_{α,β,γ}(q^{αx+γ};q^β)_{\infty}^{S_{αβγ}}}\,dx\] 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.