q-Deformation of Meromorphic Solutions of Linear Differential 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.

References in corpus (5)

• Building meromorphic solutions of $q$-difference equations using a Borel-Laplace summation
• Algebraic construction of the Stokes sheaf for irregular linear q-difference equations
• The q-analogue of the wild fundamental group (II)
• Introduction to the Galois Theory of Linear Differential Equations
• Galois theory of q-difference equations

Cited by in corpus (1)

• Adjoint Difference Equation for a Nikiforov-Uvarov-Suslov difference equation of hypergeometric type on Non-uniform Lattices