The Stokes phenomenon for the $q$-difference equation satisfied by the basic hypergeometric series ${}_3\varphi_1(a_1,a_2,a_3;b_1;q,x)$

TL;DR

This paper investigates the Stokes phenomenon for a specific basic hypergeometric series, deriving a connection formula between its behaviors near zero and infinity using q-Borel-Laplace transformations, and explores its limit as q approaches 1.

Contribution

It introduces a new connection formula for the ${}_3 ext{phi}_1$ series using q-Borel-Laplace transformations and analyzes its limit as q approaches 1.

Findings

Derived the connection formula between origin and infinity for the series

Applied q-Borel-Laplace transformations to hypergeometric series

Analyzed the limit of the connection formula as q approaches 1

Abstract

We show the connection formula for the basic hypergeometric series ${}_3\varphi_1(a_1,a_2,a_3;b_1;q,x)$ between around the origin and infinity by the using of the $q$-Borel-Laplace transformations. We also show the limit $q\to 1-0$ of the new connection formula.