Nevanlinna theory of the Hahn difference operators and its applications

TL;DR

The paper develops a Nevanlinna theory framework for the Hahn difference operator $\mathcal{D}_{q,c}$ acting on zero-order meromorphic functions, including a logarithmic derivative lemma and a second main theorem adapted to the Hahn setting. It derives defect relations, Picard-type results, and a Five-Value theorem, extending classical value-distribution results to Hahn differences. The theory is then applied to linear Hahn-difference equations and Fermat-type Hahn-difference equations, yielding growth restrictions and nonexistence results for finite-order meromorphic solutions. The work unifies $q$-Jackson and forward differences within Nevanlinna theory and highlights open questions, such as a potential Hahn-difference Wiman-Valiron theory. These results enrich discrete complex analysis and the study of Hahn-difference dynamical systems.

Abstract

This paper establishes the version of Nevanlinna theory based on Hahn difference operator $\mathcal{D}_{q,c}(g)=\frac{g(qz+c)-g(z)}{(q-1)z+c}$ for meromorphic function of zero order in the complex plane $\mathbb{C}$. We first establish the logarithmic derivative lemma and the second fundamental theorem for the Hahn difference operator. Furthermore, the deficiency relation, Picard's theorem and the five-value theorem are extended to the setting of Hahn difference operators by applying the second fundamental theorem. Finally, we also consider the solutions of complex linear Hahn difference equations and Fermat type Hahn difference equations.

Theorems & Definitions (23)

• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4