Nevanlinna theory of the Hahn difference operators and its applications

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.