# Nevanlinna theory for Jackson difference operators and entire solutions of q-difference equations

## TL;DR

This paper develops a version of Nevanlinna theory tailored for Jackson $q$-difference operators and applies it to analyze the growth and solutions of linear $q$-difference equations with meromorphic coefficients.

## Contribution

It introduces a Nevanlinna theory framework based on Jackson $q$-difference operators and applies it to study entire solutions of linear $q$-difference equations.

## Key findings

- Established a Jackson $q$-difference version of the logarithmic difference lemma
- Derived the second fundamental theorem and Picard theorem for Jackson $q$-difference operators
- Estimated the logarithmic order of certain $q$-special functions

## Abstract

This paper establishes a version of Nevanlinna theory based on Jackson difference operator $D_{q}f(z)=\frac{f(qz)-f(z)}{qz-z}$ for meromorphic functions of zero order in the complex plane $\mathbb{C}$. We give the logarithmic difference lemma, the second fundamental theorem, the defect relation, Picard theorem and five-value theorem in sense of Jackson $q$-difference operator. By using this theory, we investigate the growth of entire solutions of linear Jackson $q$-difference equations $D^{k}_{q}f(z)+A(z)f(z)=0$ with meromorphic coefficient $A,$ where $D^k_q$ is Jackson $k$-th order difference operator, and estimate the logarithmic order of some $q$-special functions.