On the meromorphic solutions of some linear difference equations
- PDF / 232,579 Bytes
- 12 Pages / 595.28 x 793.7 pts Page_size
- 21 Downloads / 238 Views
RESEARCH
Open Access
On the meromorphic solutions of some linear difference equations Huifang Liu1* and Zhiqiang Mao2* *
Correspondence: [email protected]; [email protected] 1 Institute of Mathematics and Informatics, Jiangxi Normal University, Nanchang, 330022, China 2 School of Mathematics and Computer, Jiangxi Science and Technology Normal University, Nanchang, 330038, China
Abstract This paper is devoted to studying the growth of meromorphic solutions of some linear difference equations. We obtain some results on the growth of meromorphic solutions when most coefficients in such equations have the same order, which are supplements of previous results due to Chiang and Feng, and Laine and Yang. Some examples are given to show the sharpness of our results. MSC: 39B32; 30D35; 39A10 Keywords: complex difference equation; meromorphic solution; growth
1 Introduction and main results In this paper, the term meromorphic function will mean meromorphic in the whole complex plane C. It is assumed that the reader is familiar with the standard notations and basic results of Nevanlinna theory (see, e.g., [–]). In addition, we use σ (f ) and σ (f ) to denote the order and the hyper-order of a meromorphic function f (z), and λ(f ) and λ(/f ) to denote the exponents of convergence of zeros and poles of f (z), respectively. For a meromorphic function f (z), when < σ (f ) < ∞ or < σ (f ) < ∞, its type τ (f ) and hyper-type ) ) and τ (f ) = limr→∞ logrσT(r,f (see, e.g., [, , ]). τ (f ) are defined by τ (f ) = limr→∞ T(r,f (f ) rσ (f ) Recently, meromorphic solutions of complex difference equations have become a subject of great interest from the viewpoint of Nevanlinna theory, due to the apparent role of the existence of such solutions of finite order for the integrability of discrete difference equations (see, e.g., [–]). Halburd and Korhonen [] proved that when the difference equation ω(z + ) + ω(z – ) = R(z, ω),
(.)
where R(z, ω) is rational in both of its arguments and has an admissible meromorphic solution of finite order, then either ω satisfies a difference Riccati equation, or equation (.) can be transformed by a linear change in ω(z) to a difference Painlevé equation or a linear difference equation. Thus the linear difference equation plays an important role in the study of properties of difference equations. Chiang and Feng [] considered the linear difference equation ak (z)f (z + k) + · · · + a (z)f (z + ) + a (z)f (z) = ,
(.)
and obtained the following results. © 2013 Liu and Mao; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Liu and Mao Advances in Difference Equations 2013, 2013:133 http://www.advancesindifferenceequations.com/content/2013/1/133
Theorem A [] Let a (z), . . . , ak (z) be polynomials. If there exists an integer l ( ≤ l ≤ k)
Data Loading...
