Spectral types of linear $q$-difference equations and $q$-analog of middle convolution

TL;DR

This paper introduces a $q$-analog of middle convolution for linear $q$-difference equations, preserving key properties and relating to Euler transformations, extending Katz's differential case to the $q$-difference setting.

Contribution

It defines a $q$-analog of middle convolution, demonstrating its properties and relation to Euler transformation for $q$-difference equations.

Findings

Transforms Fuchsian type equations to Fuchsian type equations

Preserves rigidity index of $q$-difference equations

Expressed as a $q$-analog of Euler transformation

Abstract

We give a $q$-analog of middle convolution for linear $q$-difference equations with rational coefficients. In the differential case, middle convolution is defined by Katz, and he examined properties of middle convolution in detail. In this paper, we define a $q$-analog of middle convolution. Moreover, we show that it also can be expressed as a $q$-analog of Euler transformation. The $q$-middle convolution transforms Fuchsian type equation to Fuchsian type equation and preserves rigidity index of $q$-difference equations.