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

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 $D_q^{k}f(z)+A(z)f(z)=0$ with meromorphic coefficient $A,$ where $D_q^k$ is Jackson $k$-th order difference operator, and estimate the logarithmic order of some $q$-special functions.

25 pages. We give some modifications containing typing errors

References in corpus (5)

• Nevanlinna theory for the difference operator
• Logarithmic difference lemma in several complex variables and partial difference equations
• On the $q$-partial differential equations and $q$-series
• Nevanlinna Theory of the Wilson Divided-difference Operator
• Wiman-Valiron theory for a polynomial series based on the Wilson operator