Results on the growth of meromorphic solutions of some linear difference equations with meromorphic coefficients
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RESEARCH
Open Access
Results on the growth of meromorphic solutions of some linear difference equations with meromorphic coefficients Zhang Li Yuan and Qiu Ling* *
Correspondence: [email protected] College of Applied Science, Beijing University of Technology, Beijing, 100124, China
Abstract In this paper, we investigate the growth of meromorphic solutions of some linear difference equations. We obtain some new results on the growth of meromorphic solutions when most coefficients in such equations are meromorphic functions, which are supplements of previous results due to Li and Chen (Adv. Differ. Equ. 2012:203, 2012) and Liu and Mao (Adv. Differ. Equ. 2013:133, 2013). MSC: 30D35; 39A10 Keywords: complex difference equation; meromorphic coefficients; growth
1 Introduction and main results In this article, a meromorphic function always means meromorphic in the whole complex plane C, and c always means a nonzero constant. We adopt the standard notations of the Nevanlinna value distribution theory of meromorphic functions such as T(r, f ), m(r, f ) and N(r, f ) as explained in [–]. In addition, we will use notations ρ(f ) to denote the order of growth of a meromorphic function f (z), λ(f ) to denote the exponents of convergence of the zero sequence of a meromorphic function f (z), λ( f ) to denote the exponents of convergence of the pole sequence of a meromorphic function f (z), and we define them as follows: ρ(f ) = lim sup r→∞
λ(f ) = lim inf r→∞
log T(r, f ) , log r
log N(r, f ) log r
,
log N(r, f ) . = lim inf λ r→∞ f log r 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. About the growth of meromorphic solutions of some linear difference equations, some results can be found in [–]. Laine and Yang [] considered the entire functions coefficients case and got the following. © 2014 Yuan and Ling; 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.
Yuan and Ling Advances in Difference Equations 2014, 2014:306 http://www.advancesindifferenceequations.com/content/2014/1/306
Theorem A [] Let Aj (z) (≡ ) (j = , , . . . , n) be entire functions of finite order such that among those coefficients having the maximal order ρ := max≤j≤n ρ(Aj ), exactly one has its type strictly greater than the others. Then, for any meromorphic solution f (z) to An (z)f (z + n) + · · · + A (z)f (z + ) + A (z)f (z) = , we have ρ(f ) ≥ ρ + . Chiang and Feng [, ] improved Theorem A as follows. Theorem B [, ] Let Aj (z) (≡ ) (j = , , . . . , n) be entire functions such that there exists an integer l, ≤ l ≤ n, such that ρ(Al ) > max ρ(Aj ). ≤j≤n,j=l
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