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arXiv · 1812.10014

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

Tingbin Cao·Huixin Dai·Jun Wang
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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.

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Tingbin Cao, Huixin Dai, Jun Wang. 2019-09-04. Nevanlinna theory for Jackson difference operators and entire solutions of q-difference equations. https://doi.org/10.1007/s10476-021-0092-8

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