  In this paper, we use the Mellin-Barnes-Watson method to relate solutions of
a certain type of $q$-difference equations at $Q=0$ and $Q=\infty$. We consider
two special cases; the first is the $q$-difference equation of $K$-theoretic
$I$-function of the quintic, which is degree 25; we use Adams' method to find
the extra 20 solutions at $Q=0$. The second special case is a fuchsian case,
which is confluent to the differential equation of the cohomological
$I$-function of the quintic. We compute the connection matrix and study the
confluence of the $q$-difference structure.
