q-Deformation of Meromorphic Solutions of Linear Differential Equations

TL;DR

This paper studies the behavior of meromorphic solutions of linear q-difference equations as q approaches 1, showing convergence to solutions of differential equations and constructing q-deformations of given differential equations.

Contribution

It establishes conditions under which meromorphic solutions of q-difference equations converge to those of differential equations and provides a method to construct q-deformations of differential equations.

Findings

Meromorphic solutions of q-difference equations converge to differential solutions as q approaches 1.

Construction of q-deformations for differential equations with specific Newton polygon properties.

A framework for discretizing differential equations via q-difference equations.

Abstract

In this paper, we consider the behaviour, 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.