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

We prove a connection formula for the basic hypergeomtric function ${}_n{\phi }_{n-1}(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_{n-1},0;b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_{n-1};q,z)$ 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 $q\to 1$, the $q$-Stokes matrices recover the known expressions of the Stokes matrices of the corresponding confluent hypergeometric differential system.