Confluence de solutions méromorphes d’équations aux q-différences

International audienceIn 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 summa-tion 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 hy-pergeometric series. Following Sauloy, we prove how a basis of solutions of a linear differential equation can be uniformly approximated on a convenient domain by a basis 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