  We study the divergent basic hypergeometric series which is a $q$-analog of
divergent hypergeometric series. This series formally satisfies the linear
$q$-difference equation. In this paper, for that equation, we give an actual
solution which admits basic hypergeometric series as a $q$-Gevrey asymptotic
expansion. Such an actual solution is obtained by using $q$-Borel summability,
which is a $q$-analog of Borel summability. Our result shows a $q$-analog of
the Stokes phenomenon. Additionally, we show that letting $q\to1$ in our result
gives the Borel sum of classical hypergeometric series. The same problem was
already considered by Dreyfus, but we note that our result is remarkably
different from his one.
