Nevanlinna Theory for Jackson Difference Operators and Entire Solutions of q-Difference Equations

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