Explicit evaluation of the $q$-Stokes matrices for certain confluent hypergeometric $q$-difference equations

TL;DR

This paper derives explicit formulas for the $q$-Stokes matrices of certain confluent hypergeometric $q$-difference equations, connecting them to classical Stokes matrices as $q$ approaches 1, using $q$-Borel resummation.

Contribution

It provides the first explicit computation of $q$-Stokes matrices for specific confluent hypergeometric $q$-difference systems, linking $q$-difference and differential equations.

Findings

Explicit $q$-Stokes matrices computed for a class of confluent hypergeometric $q$-difference equations.

Demonstrated that $q$-Stokes matrices recover classical Stokes matrices as $q o 1$.

Applied $q$-Borel resummation to derive connection formulas.

Abstract

We prove a connection formula for the basic hypergeomtric function ${}_n\varphi_{n-1}\left( a_1,...,a_{n-1},0; b_1,...,b_{n-1} ; q, z\right)$ by using the $q$-Borel resummation. As an application, we compute $q$-Stokes matrices of a special confluent hypergeometric $q$-difference system with an irregular singularity. We show that by letting $q\rightarrow 1$, the $q$-Stokes matrices recover the known expressions of the Stokes matrices of the corresponding confluent hypergeometric differential system.