Abstract
In a recent paper [1], Ablowitz, Halburd and Herbst 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 degy R(z, y) ≤ 2. Improvements and extensions of such results are presented in this paper. In addition to order considerations, a result (see Theorem 13) is proved to indicate that solutions having Borel exceptional zeros and poles seem to appear in special situations only.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. J. Ablowitz, R. Halburd, and B. Herbst, On the extension of the Painlevé property to difference equations, Nonlinearity 13 (2000), 889–905.
S. Bank and R. Kaufman, An extension of Hölder’s theorem concerning the gamma function, Funkcialaj Ekvacioj 19 (1976), 53–63.
L. Carleson and T. Gamelin, Complex Dynamics, Springer-Verlag, New York, 1993.
J. Clunie, The composition of entire and meromorphic functions, 1970 Mathematical Essays Dedicated to A. J. Macintyre, Ohio University Press, Athens, Ohio, 75–92.
G. Gundersen, J. Heittokangas, I. Laine, J. Rieppo and D. Yang, Meromorphic solutions of generalized Schröder equations, Aequationes Math. 63 (2002), 110–135.
W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
G. Jank and L. Volkmann, Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, Birkhäuser Verlag, Basel-Boston, 1985.
I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993.
S. Shimomura, Entire solutions of a polynomial difference equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 253–266.
N. Yanagihara, Meromorphic solutions of some difference equations, Funkcialaj Ekvacioj 23 (1980), 309–326.
Author information
Authors and Affiliations
Corresponding author
Additional information
J.H., R.K., I.L., and J.R. have been partially supported by the INTAS project grant 99-00089. K.T. has been partially supported by the Academy of Finland and the JSPS (the Grants-in-Aid for Scientific Research System, No. 12740085).
Rights and permissions
About this article
Cite this article
Heittokangas, J., Korhonen, R., Laine, I. et al. Complex Difference Equations of Malmquist Type. Comput. Methods Funct. Theory 1, 27–39 (2001). https://doi.org/10.1007/BF03320974
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03320974