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
