Confluence of meromorphic solutions of q-difference equations

Confluence de solutions méromorphes d’équations aux q-différences

Thomas Dreyfus (1)

1 IMJ - Institut de Mathématiques de Jussieu

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 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.

Domaines: Mathématiques générales [math.GM]

Thomas Dreyfus. Confluence of meromorphic solutions of q-difference equations. Annales de l'Institut Fourier, 2015, 65 (2), pp.431 - 507. ⟨10.5802/aif.2937⟩. ⟨hal-01897298⟩