Abstract

In this paper, we prove that for a transcendental entire function \(f\) of finite order such that \(\lambda(f-a)<\rho(f)\), where \(a\) is an entire function and satisfies \(\rho(a)<\rho(f)\), \(n\in\mathbb{N}\), if \(\Delta_{c}^{n}f\) and \(f\) share the entire function \(b\) satisfying \(\rho(b)<\rho(f)\) CM, where \(c\in\mathbb{C}\) satisfies \(\Delta_{c}^{n}f\not\equiv 0\), then \(f(z)=a(z)+de^{cz}\), where \(d,c\) are two nonzero constants. In particular, if \(a=b\), then \(a\) reduces to a constant. This result improves and generalizes the recent results of Chen and Chen [3], Liao and Zhang [10] and Lü et al. [11] in a large scale. Also we exhibit some relevant examples to fortify our results.

中文翻译：

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整个函数及其高阶差分运算符

摘要

在本文中，我们证明对于有限阶的超越整函数\(f\)使得\(\lambda(fa)<\rho(f)\)，其中\(a\)是整函数并且满足\(\rho(a)<\rho(f)\)、\(n\in\mathbb{N}\)、如果\(\Delta_{c}^{n}f\)和\(f\)共享满足\(\rho(b)<\rho(f)\) CM的整个函数\(b\)，其中\(c\in\mathbb{C}\)满足\(\Delta_{c}^{ n}f\not\equiv 0\)，然后\(f(z)=a(z)+de^{cz}\)，其中\(d,c\)是两个非零常数。特别是，如果\(a=b\)，则\(a\)减少为常数。这一结果改进并推广了Chen和Chen [3]、Liao和Zhang [10]以及Lü等人的最新结果。[11] 大规模。我们还展示了一些相关示例来强化我们的结果。