q-Deformation of meromorphic solutions of linear differential equations

In this paper, we consider the behavior, when q goes to 1, of the set of a convenient basis of meromorphic solutions of a family of linear q-difference equations. In particular, we show that, under convenient assumptions, such basis of meromorphic solutions converges, when q goes to 1, to a basis of meromorphic solutions of a linear differential equation. We also explain that given a linear differential equation of order at least two, which has a Newton polygon that has only slopes of multiplicities one, and a basis of meromorphic solutions, we may build a family of linear q-difference equations that discretizes the linear differential equation, such that a convenient family of basis of meromorphic solutions is a q-deformation of the given basis of meromorphic solutions of the linear differential equation.