Meromorphic solutions of linear $q$-difference equations

TL;DR

This paper constructs explicit meromorphic solutions for first-order linear q-difference equations, analyzes their zeros and poles, and extends the approach to higher-order equations with detailed examples.

Contribution

It provides a comprehensive method for solving and analyzing meromorphic solutions of linear q-difference equations, including higher-order cases.

Findings

Explicit solutions for first-order equations derived.

Location of zeros and poles characterized.

Extension to higher-order equations with factorization.

Abstract

In this article, we construct explicit meromorphic solutions of first order linear $q$-difference equations in the complex domain and we describe the location of all their zeros and poles. The homogeneous case leans on the study of four fundamental equations, providing the previous informations in the framework of entire or meromorphic coefficients. The inhomogeneous situation, which stems from the homogeneous one and two fundamental equations, is also described in detail. We also address the case of higher-order linear $q$-difference equations, using a classical factorization argument. All these results are illustrated by several examples.