Asymptotic formulae for Eulerian series

TL;DR

This paper derives asymptotic formulae for Eulerian series involving $q$-Pochhammer symbols, providing integral approximations and full asymptotic expansions as $q$ approaches 1 from below, with potential applications to basic hypergeometric series.

Contribution

It establishes conditions under which Eulerian series can be approximated by integrals and derives their complete asymptotic expansions near $q=1$, extending to related hypergeometric series.

Findings

Asymptotic formulae for Eulerian series as $q o 1^-$.

Integral approximations valid under monotonicity conditions.

Full asymptotic expansions for these 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 $H_{-1}(u)=vu-Au^2+\sum_{\alpha}\mathrm{li}_2(e^{-\alpha u})\sum_{\beta,\gamma} \beta^{-1}S_{\alpha\beta\gamma}$ 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.