Analytic theory of difference equations with rational and elliptic coefficients and the Riemann-Hilbert problem

TL;DR

This paper introduces a novel analytic framework for difference equations with rational and elliptic coefficients, focusing on canonical solutions, local monodromies, and isomonodromic deformations, bridging discrete and continuous monodromy concepts.

Contribution

It develops a new approach based on canonical meromorphic solutions and introduces isomonodromic deformations for elliptic coefficient difference equations.

Findings

Canonical meromorphic solutions along 'thick paths'

Definition of local monodromies for difference equations

Construction of isomonodromic deformations with changing elliptic periods

Abstract

A new approach to the analytic theory of difference equations with rational and elliptic coefficients is proposed. It is based on the construction of canonical meromorphic solutions which are analytical along "thick paths". The concept of such solutions leads to a notion of local monodromies of difference equations. It is shown that in the continuous limit they converge to the monodromy matrices of differential equations. New type of isomonodromic deformations of difference equations with elliptic coefficients changing the periods of elliptic curves is constructed.