  For transcendental functions that solve non-linear $q$-difference equations,
the best descriptions available are the ones obtained by expansion near
critical points at the origin and infinity. We describe such solutions of a
$q$-discrete Painlev\'e equation, with 7 parameters whose initial value space
is a rational surface of type $A_1^{(1)}$. The resultant expansions are shown
to approach series expansions of the classical sixth Painlev\'e equation in the
continuum limit.
