On zeros and growth of solutions of complex difference equations

[摘要] Let f be an entire function of finite order, let$n\geq 1$ ,$m\geq 1$ ,$L(z,f)\not \equiv 0$ be a linear difference polynomial of f with small meromorphic coefficients, and$P_{d}(z,f)\not \equiv 0$ be a difference polynomial in f of degree$d\leq n-1$ with small meromorphic coefficients. We consider the growth and zeros of$f^{n}(z)L^{m}(z,f)+P_{d}(z,f)$ . And some counterexamples are given to show that Theorem 3.1 proved by I. Laine (J. Math. Anal. Appl. 469:808–826, 2019) is not valid. In addition, we study meromorphic solutions to the difference equation of type$f^{n}(z)+P_{d}(z,f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z}$ , where$n\geq 2$ ,$P_{d}(z,f)\not \equiv 0$ is a difference polynomial in f of degree$d\leq n-2$ with small mromorphic coefficients,$p_{i}$ ,$\alpha _{i}$ ( $i=1,2$ ) are nonzero constants such that$\alpha _{1}\neq \alpha _{2}$ . Our results are improvements and complements of Laine 2019, Latreuch 2017, Liu and Mao 2018.

[学科分类] 航空航天科学

[关键词] Difference equations;Growth;Zeros;Meromorphic solution