On Picard exceptional values of difference polynomials

Abstract

Let f denote a transcendental meromorphic function of finite order, we have shown that the difference polynomial $P\left(f\right)$ of the form:

$f^n\left(\Delta ,f\right){}^{n_1}\left(\Delta ,{}^2,f\right){}^{n_2}\cdots \left(\Delta ,{}^k,f\right){}^{n_k}-\alpha$

cannot have 0 as its Picard exceptional value, where $\alpha$ is a small function of f. As a result of the studies, we are able to provide a unified and simplified proof to several known results, particularly to those Hayman-type of problems and conjecture, with generalizations and extensions.

Keywords::

• entire function
• Nevanlinna theory
• Picard exceptional value
• differential-difference polynomial

Keywords::

• 30D35

Acknowledgements

The authors would like to thank the referee for his/her several important suggestions and for pointing out some errors in our original manuscript. These comments greatly improved the readability of the paper.

Notes

^1. Email: [email protected]

Additional information

Funding

This works was supported by the research funds of China University of Petroleum [grant No: 14CX06085A].