# zbMATH — the first resource for mathematics

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 $y(z+ 1)+ y(z- 1)= R(z, y)$ with $$R(z, y)$$ rational in both arguments admits a transcendental meromorphic solution of finite order, then $$\deg_yR(z,y)\leq 2$$.
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 to ordinary differential equations in the complex domain 39A12 Discrete version of topics in analysis 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
Full Text:
##### 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. · Zbl 0956.39003 · 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. · Zbl 0346.33001 [3] L. Carleson and T. Gamelin, Complex Dynamics, Springer-Verlag, New York, 1993. · Zbl 0782.30022 [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. · Zbl 0218.30032 [5] G. Gundersen, J. Heittokangas, I. Laine, J. Rieppo and D. Yang, Meromorphic solutions of generalized Schröder equations, Aequationes Math. 63 (2002), 110–135. · Zbl 1012.30016 · 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. · Zbl 0682.30001 [8] I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993. · Zbl 0784.30002 [9] S. Shimomura, Entire solutions of a polynomial difference equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 253–266. · Zbl 0469.30021 [10] N. Yanagihara, Meromorphic solutions of some difference equations, Funkcialaj Ekvacioj 23 (1980), 309–326. · Zbl 0474.30024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.