Confluence of meromorphic solutions of q-difference equations

TL;DR

This paper introduces a q-analogue of Borel-Laplace summation, demonstrating how solutions of differential equations can be approximated by solutions of q-difference equations, with applications to hypergeometric series and monodromy matrices.

Contribution

It develops a novel q-analogue of Borel-Laplace summation and shows how to approximate differential equation solutions with q-difference solutions, extending the theory of Stokes and monodromy matrices.

Findings

Uniform approximation of divergent series by meromorphic solutions

Approximation of Stokes and monodromy matrices using q-invariant matrices

Application to basic hypergeometric series

Abstract

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 J. Sauloy, we prove how a fundamental set of solutions of a linear differential equation can be uniformly approximated on a convenient domain by a fundamental set 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.