有限级亚纯函数的导数与其差分导数组合分担多项式的问题

The Problem of Polynomial Sharing between the Derivative and Its Differential-Difference Combination of Finite-Order Meromorphic Functions

摘要: 本文研究了有限级亚纯函数$f\left(z\right)$ 的差分导数组合$f\left(z+c\right)+f\left(z\right)+f^{\prime }\left(z\right)$ 与导数$f^{\prime }\left(z\right)$ 分担多项式的唯一性问题。基于Nevanlinna值分布理论，我们证明了：若$f\left(z+c\right)+f\left(z\right)+f^{\prime }\left(z\right)$ 与$f^{\prime }\left(z\right)$ 分担两个互异多项式及无穷远点，函数$f\left(z\right)$ 则满足以下两种情形之一：(1) $f\left(z\right)=f\left(z+2c\right)$ ；(2) $f\left(z\right)$ 退化为多项式，若存在非零常数$A\left(\ne 1\right)$ 与常系数多项式$P\left(z\right)$ ，则恒等式$f^{\prime }\left(z\right)-A\left(f\left(z+c\right)+f\left(z\right)+f^{\prime }\left(z\right)\right)\equiv \left(1-A\right)P\left(z\right)$ 成立。所得结果进一步推广了亚纯函数差分算子分担多项式的唯一性理论。

Abstract: This paper studies the uniqueness problem that the differential-difference combination $f\left(z+c\right)+f\left(z\right)+f^{\prime }\left(z\right)$ and the derivative $f^{\prime }\left(z\right)$ of finite-order meromorphic function $f\left(z\right)$ share polynomials. By using Nevanlinna value distribution theory, it is proved that if $f\left(z+c\right)+f\left(z\right)+f^{\prime }\left(z\right)$ and $f^{\prime }\left(z\right)$ share two distinct polynomials and infinity, then $f\left(z\right)$ satisfies one of the following two cases: 1) $f\left(z\right)=f\left(z+2c\right)$ ; 2) $f\left(z\right)$ is a polynomial, and there exists a non-zero constant $A\ne 1$ and a polynomial $P\left(z\right)$ with constant coefficients such that $f^{\prime }\left(z\right)-A\left(f\left(z+c\right)+f\left(z\right)+f^{\prime }\left(z\right)\right)\equiv \left(1-A\right)P\left(z\right)$ . The results obtained generalize the uniqueness theory of meromorphic functions concerning difference operators sharing polynomials.