  In this paper, we consider a q-analogue of the Borel-Laplace summation where
q>1 is a real parameter. In particular, we show that the Borel-Laplace
summation of a divergent power series solution of a linear differential
equation can be uniformly approximated on a convenient sector, by a meromorphic
solution of a corresponding family of linear q-difference equations. We perform
the computations for the basic hypergeometric series. Following J. Sauloy, we
prove how a fundamental set of solutions of a linear differential equation can
be uniformly approximated on a convenient domain by a fundamental set of
solutions of a corresponding family of linear q-difference equations. This
leads us to the approximations of Stokes matrices and monodromy matrices of the
linear differential equation by matrices with entries that are invariants by
the multiplication by q.
