Derivatives of Meromorphic Functions Sharing Polynomials with Their Difference Operators

Pub Date : 2023-12-28 DOI:10.3103/s1068362323060079

M.-H. Wang, J.-F. Chen

引用次数: 0

Abstract

In this paper, we investigate the uniqueness of meromorphic functions of finite order \(f(z)\) concerning their difference operators \(\Delta_{c}f(z)\) and derivatives \(f^{\prime}(z)\) and prove that if \(\Delta_{c}f(z)\) and \(f^{\prime}(z)\) share \(a(z)\), \(b(z)\), \(\infty\) CM, where \(a(z)\) and \(b(z)\) are two distinct polynomials, then they assume one of following cases: \((1)\) \(f^{\prime}(z)\equiv\Delta_{c}f(z)\); \((2)\) \(f(z)\) reduces to a polynomial and \(f^{\prime}(z)-A\Delta_{c}f(z)\equiv(1-A)(c_{n}z^{n}+c_{n-1}z^{n-1}+\cdots+c_{1}z+c_{0})\), where \(A(\neq 1)\) is a nonzero constant and \(c_{n},c_{n-1},\cdots,c_{1},c_{0}\) are all constants. This generalizes the corresponding results due to Qi et al. and Deng et al.

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与它们的差分算子共享多项式的单项式函数的衍生物

Abstract 在本文中，我们研究了有限阶微变函数 \(f(z)\) 关于其差分算子 \(\Delta_{c}f(z)\) 和导数 \(f^{\prime}(z)\) 的唯一性，并证明如果 \(\Delta_{c}f(z)\) 和 \(f^{\prime}(z)\) 共享 \(a(z)\)、\CM, 其中（a(z)）和（b(z)）是两个不同的多项式，那么它们假设以下情况之一：\((1)\)\(f^{\prime}(z)\equiv\Delta_{c}f(z)\);\((2)/)（f(z)）减为多项式，并且（f^{\prime}(z)-A\Delta_{c}f(z)\equiv(1-A)(c_{n}z^{n}+c_{n-1}z^{n-1}+\cdots+c_{1}z+c_{0}）、其中 \(A(\neq 1)\)是一个非零常数，而 \(c_{n},c_{n-1},\cdots,c_{1},c_{0}\) 都是常数。这概括了 Qi 等人和 Deng 等人的相应结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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