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.
Categories: math.CA