An asymptotic formula of the divergent bilateral basic hypergeometric series

TL;DR

This paper derives an asymptotic formula for a divergent bilateral basic hypergeometric series using the q-Borel-Laplace method and examines its limit as q approaches 1 from below.

Contribution

It introduces a new asymptotic formula for a specific divergent series and analyzes its behavior as q approaches 1, expanding understanding of basic hypergeometric series.

Findings

Derived an asymptotic formula for ${}_1 ext{psi}_0$ series

Applied q-Borel-Laplace method to divergent series

Analyzed the limit as q approaches 1 from below

Abstract

We show an asymptotic formula of the divergent bilateral basic hypergeometric series ${}_1\psi_0 (a;-;q,\cdot)$ with using the $q$-Borel-Laplace method. We also give the limit $q\to 1-0$ of our asymptotic formula.