By Cartan’s version of Nevanlinna’s theory, we prove the following result: Let and be two positive integers satisfying and , let be a polynomial such that , let be a finite complex number such that and , let be distinct finite nonzero complex numbers, and let be either an exponential polynomial of degree less than or an ordinary polynomial in such that for . Suppose that is a nonconstant meromorphic solution of the difference equation
Then and is reduced to a transcendental entire function such that its order or its order satisfies with , while can be either expressed as
where and are entire function such that their characteristic functions satisfy
as and . Here, is a subset of finite linear measure.
An example is provided to show that the assumption , in a sense, is the best possible.
This result improves and extends the corresponding results from Yang–Laine (2010), Liu–Lü–Shen–Yang (2014), and Latreuch (2017).
"On the existence of meromorphic solutions of certain nonlinear difference equations." Rocky Mountain J. Math. 51 (5) 1723 - 1748, October 2021. https://doi.org/10.1216/rmj.2021.51.1723