Confluence of meromorphic solutions of $q$-difference equations

Dreyfus, Thomas

Annales de l'Institut Fourier, Tome 65 (2015) no. 2, pp. 431-507.

Dans cet article, nous considérons un $q$-analogue du processus de sommation de Borel-Laplace, avec $q>1$ paramètre réel. En particulier, nous prouvons que la sommation de Borel-Laplace d’une série formelle solution d’une équation différentielle linéaire peut être approchée, dans un secteur convenable, par une solution méromorphe d’une certaine famille d’équations aux $q$-différences linéaire. Nous faisons les calculs pour les séries hypergéométriques. En s’inspirant de Sauloy, nous prouvons comment une base de solutions d’une equation différentielle linéaire peut être approchée, sur un secteur convenable, par une base de solutions d’une famille correspondante d’équations aux $q$-différences. Cela nous mène à l’approximation des matrices de Stokes et de monodromies de l’équation différentielle, par des matrices dont les entrées sont invariantes par multiplication par $q$.

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 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 $q$.

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