Asymptotic formulae for Eulerian series

Abstract

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 $H_{-1}(u)=vu-A{u}^2+{\sum }_{\alpha }{\mathrm{li}}_2(e^{-\alpha u}){\sum }_{\beta ,\gamma }{\beta }^{-1}{S}_{\alpha \beta \gamma }$ and consider the Eulerien series \[\mathcal{H}(z;q)=\sum_{m=0}^{\infty}\frac{q^{Am^2+Bm}z^{m}}{\prod\limits_{α,β,γ}(q^{αm+γ};q^β)_{\infty}^{S_{αβγ}}}.\] We prove that if there exist an $\epsilon >0$ such that $H_{-1}(u)$ is an increasing function on $[0,\epsilon )$, then as $q\to 1^{-}$, \[\mathcal{H}(z;q)=\left(1+o\left(|\log q|^p\right)\right)\int\limits_{0}^{\infty}\frac{q^{Ax^2+Bx}z^{x}}{\prod\limits_{α,β,γ}(q^{αx+γ};q^β)_{\infty}^{S_{αβγ}}}\,dx\] holds for each $p\ge 0$. We also obtain full asymptotic expansions for $\mathcal{H}(z;q)$ which satisfy above condition as $q\to 1^{-}$. The complete asymptotic expansions for related basic hypergeometric series could be derived as special cases.

3^3 pages