涉及亚纯函数差分算子的唯一性定理
Uniqueness of Meromorphic Function Concerning Difference Operator
DOI: 10.12677/PM.2022.121025, PDF,  被引量    国家自然科学基金支持
作者: 邱仕林, 郑瑞林, 刘 丹*：华南农业大学数学与信息学院，广东 广州
关键词: 亚纯函数；唯一性；差分算子；Meromorphic Function； Uniqueness； Difference Operator
摘要: 本文运用Nevanlinna值分布论研究了有穷级亚纯函数与其差分算子分担函数的问题，得到了如下结果。设f(z)是有穷级超越亚纯函数，a(z)(≢∞),b(z)是f(z)的Borel例外函数且a(z),b(z)∈S(f)，其中a(z)是满足ρ(a(z)) < 1的亚纯函数。令η是满足Δηf(z)≢0的非零常数。如果f(z)和Δηf(z)CM分担Δηa(z),b(z)，那么，a(z)=0，b(z)=∞，f(z)=BeAz，其中A,B是非零常数。本文是对陈创鑫和张然然结果的改进和推广。
Abstract: In this paper, we study the uniqueness of meromorphic functions by Nevanlinna value distribution theory and obtain the following result. Let f(z) be a transcendental meromorphic function of finite order and a(z)(∈S(f)),b(z)(∈S(f)) be a Borel exceptional function of f(z), where a(z)(≢∞) is a meromorphic function satisfying ρ(a(z)) < 1. Let η be a nonzero constant satisfying Δηf(z)≢0. If f(z) and Δηf(z) share Δηa(z),b(z) CM, then a(z)=0，b(z)=∞，f(z)=BeAz, where A,B are non-zero constants. Our result is an improvement of the theorem given by Chen and Zhang.
文章引用：邱仕林, 郑瑞林, 刘丹. 涉及亚纯函数差分算子的唯一性定理[J]. 理论数学, 2022, 12(1): 209-217. https://doi.org/10.12677/PM.2022.121025

参考文献

[1] Hayman, W.K. (1964) Meromorphic Functions. Clarendon Press, Oxford.
[2] Yang, C.C. and Yi, H.X. (2003) Uniqueness Theory of Meromorphic Functions. Kluwer Academic Publishers Group, Dordrecht.
[3] Yang, L. (1993) Value Distribution Theory. Springer, Berlin.
[4] Csillag, P. (1935) Uber ganze Funktionen, welehe drei nicht verschwindende Ableitungen besitzen. Mathematische Annalen, 110, 745-752. [Google Scholar] [CrossRef] 
[5] Tumura, Y. (1937) On the Extensions of Borels Theorem and Saxer-Csillags Theorem. Proceedings of the Physico-Mathematical Society of Japan, 19, 29-35.
[6] Halburd, R.G. and Korhonen, R.J. (2006) Difference Analogue of the Lemma on the Logaritheoremic Derivative with Applications to Difference Equations. Journal of Mathematical Analysis and Applications, 314, 477-487. [Google Scholar] [CrossRef] 
[7] Halburd, R.G. and Korhonen, R.J. (2006) Nevanlinna Theory for the Difference Operator. Annales Academiae Scientiarum Fennicae Mathematica, 31, 463-478.
[8] Chiang, Y.M. and Feng, S.J. (2008) On the Nevanlinna Characteristic of and Difference Equations in the Complex Plane. The Ramanujan Journal, 16, 105-129. [Google Scholar] [CrossRef] 
[9] Chen, Z.X. (2013) Relation-ships between Entire Functions and Their Forward Differences. Complex Variables and Elliptic Equations, 58, 299-307. [Google Scholar] [CrossRef] 
[10] Chen, C.X. and Chen, Z.X. (2016) The Uniqueness of Meromorphic Functions and Their Differences. Acta Mathematica Sinica, 59, 821-834.
[11] 陈创鑫, 张然然. 涉及整函数差分算子的唯一性定理[J]. 数学年刊A辑, 2021, 42(1): 11-22.
[12] Bergweiler, W. and Langley, J.K. (2007) Zeros of Differences of Meromorphic Functions. Mathematical Proceedings of the Cambridge Philosophical Society, 142, 133-147. [Google Scholar] [CrossRef] 
[13] Fang, M.L. and Wang, Y.F. (2021) Higher Order Difference Operators and Uniqueness of Meromorphic Functions. Analysis and Mathematical Physics, 11, Article No. 93. [Google Scholar] [CrossRef] 
[14] Laine, I. (1993) Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin. [Google Scholar] [CrossRef] 
[15] Gross, F. (1972) Factorization of Meromorphic Function. U.S. Government Printing Office, Washington DC.
[16] Laine, I. and Yang, C.C. (2007) Clunie Theorems for Difference and Q-Difference Polynomials. Journal of the London Mathematical Society, 76, 556-566. [Google Scholar] [CrossRef] 

为你推荐