The Stokes phenomenon for linear q-difference equations

After a bare minimum of theoretical background, we present two examples of the Stokes phenomenon for linear analytic q-difference equations. The first example is a q-analog of the Euler series, the so-called Tshakaloff series, an obvious touchstone for any theory of the Stokes phenomenon. The second example is the square of the Tshakaloff series, because it features striking differences with the "classical" case of ordinary differential equations. In both examples, we emphasize the link for local analytic classification and we say a word of the application to Galois theory. This rests on common work with Jean-Pierre Ramis and Changgui Zhang.