# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Complex difference equations of Malmquist type. (English) Zbl 1013.39001

Summary: M. J. Ablowitz, R. Halburd and B. Herbst [Nonlinearity 13, No. 3, 889-905 (2000; Zbl 0956.39003)] applied Nevanlinna theory to prove some results on complex difference equations reminiscent of the classical Malmquist theorem in complex differential equations. A typical example of their results tells us that if a complex difference equation

$y\left(z+1\right)+y\left(z-1\right)=R\left(z,y\right)$

with $R\left(z,y\right)$ rational in both arguments admits a transcendental meromorphic solution of finite order, then ${deg}_{y}R\left(z,y\right)\le 2$.

Improvements and extensions of such results are presented in this paper. In addition to order considerations, a result is proved to indicate that solutions having Borel exceptional zeros and poles seem to appear in special situations only.

##### MSC:
 39A10 Additive difference equations 34M05 Entire and meromorphic solutions (ODE) 39A12 Discrete version of topics in analysis 30D35 Distribution of values (one complex variable); Nevanlinna theory
##### References:
 [1] M. J. Ablowitz, R. Halburd, and B. Herbst, On the extension of the Painlevé property to difference equations, Nonlinearity 13 (2000), 889–905. · doi:10.1088/0951-7715/13/3/321 [2] S. Bank and R. Kaufman, An extension of Hölder’s theorem concerning the gamma function, Funkcialaj Ekvacioj 19 (1976), 53–63. [3] L. Carleson and T. Gamelin, Complex Dynamics, Springer-Verlag, New York, 1993. [4] J. Clunie, The composition of entire and meromorphic functions, 1970 Mathematical Essays Dedicated to A. J. Macintyre, Ohio University Press, Athens, Ohio, 75–92. [5] G. Gundersen, J. Heittokangas, I. Laine, J. Rieppo and D. Yang, Meromorphic solutions of generalized Schröder equations, Aequationes Math. 63 (2002), 110–135. · doi:10.1007/s00010-002-8010-z [6] W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. [7] G. Jank and L. Volkmann, Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, Birkhäuser Verlag, Basel-Boston, 1985. [8] I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993. [9] S. Shimomura, Entire solutions of a polynomial difference equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 253–266. [10] N. Yanagihara, Meromorphic solutions of some difference equations, Funkcialaj Ekvacioj 23 (1980), 309–326.