复$q$-移位差分方程组亚纯解的性质研究
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"Some Properties of Meromorphic Solutions of Systems of Complex q-Shift Difference Equations"
这篇研究论文探讨了复$q$-移位差分方程系统中 meromorphic 解的一些特性。作者 Hong-Yan Xu、Bing-Xiang Liu 和 Ke-Zong Tang 来自景德镇陶瓷学院的信息与工程系。该论文于2012年9月提交，2013年3月被接受，并在Hindawi Publishing Corporation的《Abstract and Applied Analysis》期刊上发表，遵循Creative Commons Attribution License协议。
差分方程是数学中的一个重要领域，它研究的是函数及其导数或差商之间的关系。$q$-移位差分方程是一种特殊的差分方程，其中变量的阶跃不是通常的1，而是由$q$决定的一个非零常数。在这种情况下，复杂$q$-移位差分方程涉及到复数域内的函数和$q$-移位运算。
Nevanlinna 理论是研究解析函数（如meromorphic函数）的重要工具，它在理解函数的性质，特别是在值分布理论中扮演着核心角色。Meromorphic 函数是除了有限个点外全纯的函数，它们在复平面上有理函数的形式，并且可能有无穷多个极点。
文章的主要结果集中在改进和扩展了 Gao 在先前工作中的定理，关于这类方程系统的 meromorphic 解的性质。在介绍部分，作者提出了研究的目标，即深入探究复$q$-移位差分方程的 meromorphic 解的一些特定性质。这些性质可能包括解的生长行为、极点的分布以及与Nevanlinna理论的关系。
在论文的主体部分，作者可能通过分析解的级数表示、运用Nevanlinna定理的关键工具，如第一基本引理、第二基本引理等，来证明一系列关于解的性质。这些证明可能涉及解的指数增长、极点的密度和周期性等特性，从而揭示这些解的结构和行为。
具体来说，他们可能证明了某些解的存在性和唯一性，或者给出了限制解的增长速度的条件。此外，还可能讨论了解的零点和极点之间的关系，以及这些关系如何影响整体解的性质。这些结果对于理解和预测复$q$-移位差分方程系统的行为至关重要，对理论数学和应用数学，特别是动力系统、数论和控制理论等领域具有潜在的应用价值。
由于篇幅所限，文章的具体定理和证明未能在此详细展开，但可以确定的是，该论文为复$q$-移位差分方程的研究提供了新的见解和理论贡献，深化了我们对这一领域meromorphic解的理解。
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 680956, 6 pages
http://dx.doi.org/10.1155/2013/680956
Research Article
Some Properties of Meromorphic Solutions of Systems of
Complex q-Shift Difference Equations
Hong-Yan Xu, Bing-Xiang Liu, and Ke-Zong Tang
Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi 333403, China
Correspondence should be addressed to Hong-Yan Xu; xhyhhh@126.com
Received 2 September 2012; Accepted 29 March 2013
Academic Editor: Fasma Diele
Copyright © 2013 Hong-Yan Xu et al. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In view of Nevanlinna theory, we study the properties of meromorphic solutions of systems of a class of complex dierence
equations. Some results obtained improve and extend the previous theorems given by Gao.
1. Introduction and Main Results
epurposeofthispaperistostudysomepropertiesof
meromorphic solutions of complex -shi dierence equa-
tions. e fundamental results and the standard notations
of the Nevanlinna value distribution theory of meromorphic
functions will be used (see [1–3]). Besides, for meromorphic
function , a meromorphic function () is called small
function with respect to if (,()) = ((,)) =
(,)for all outside a possible exceptional set of nite
logarithmic measure lim
𝑟→∞
∫
[1,𝑟)∩𝐸
(/)<∞.
In recent years, it has been a heated topic to study dif-
ference equations, dierence product, and -dierence in the
complex plane C. ere were articles focusing on the growth
of solutions of dierence equations, value distribution and
uniqueness of dierences analogues of Nevanlinna’s theory
(see [4–9]). Chiang and Feng [10] and Halburd and Korhonen
[11]establishedadierenceanalogueofthelogarithmic
derivative lemma independently, and Barnett et al. [5]also
established an analogue of the logarithmic derivative lemma
on -dierence operators. By applying these theorems, a
number of results on meromorphic solutions of complex
dierence and -dierence equations were obtained (see [12–
19]).
In 2011, Korhonen [20] investigated the properties of
nite-order meromorphic solution of the equation
(
,
)
(
,
)
=
(
,
)
,
(1)
where (,) = (,(),(+
1
),...,(+
𝑛
)),
1
,...,
𝑛
∈C and obtained the following result.
eorem 1 (see [20]). Let () be a nite-order meromorphic
solution of (1),where(,)is a homogeneous dierence poly-
nomial with meromorphic coecients and (,)and (,)
are polynomials in ()with meromorphic coecients having
no common factors. If max{deg
𝜔
(),deg
𝜔
()−deg
𝜔
()} >
min{deg
𝜔
(), ord
0
()} − ord
0
(),then(,) =(,),
where ord
0
() denotes the order of zero of (,
0
,
1
,...,
𝑛
)
at
0
=0with respect to the variable
0
.
Let
𝑗
∈C for =1,...,, and let be a nite set of multi-
indexes =(
0
,...,
𝑛
). en a dierence polynomial of a
meromorphic function () is dened as
(
,
)
=,
(
)
,+
1
,...,+
𝑛
=
𝜆∈𝐼
𝜆
(
)
(
)
𝜆
0
+
1
𝜆
1
⋅⋅⋅+
𝑛
𝜆
𝑛
,
(2)
where the coecients
𝜆
() are small with respect to ()
in the sense that (,
𝜆
) = ((,))as tends to innity
outside of an exceptional set of nite logarithmic measure.
At the same year, Zheng and Chen [21] consider the value
distribution of meromorphic solutions of zero order of a kind
of -dierence equations and obtained the following result
which is an extension of eorem 1.
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