Abstract
Halburd and Korhonen have shown that the existence of sufficiently many meromorphic solutions of finite order is enough to single out a discrete form of the second Painlevé equation from a more general class f (z + 1) + f (z − 1) = R(z, f) of complex difference equations. A key lemma in their reasoning is to show that f (z) has to be of infinite order, provided that degf R(z, f) ≤ 2 and that a certain growth condition for the counting function of distinct poles of f (z) holds. In this paper, we prove a generalization of this lemma to higher order difference equations of more general type. We also consider related complex functional equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Ablowitz, R. Halburd and B. Herbst, On the extension of the Painlevé property to difference equations, Nonlinearity 13 (2000), 889–905.
P. M. Cohn, Algebra, Vol. 1. John Wiley & Sons, London - New York - Sydney, (1974).
R. Goldstein, Some results on factorisation of meromorphic functions, J. London Math. Soc. (2) 4 (1971), 357–364.
R. Goldstein, On meromorphic solutions of certain functional equations, Aequationes Math. 18 (1978), 112–157.
B. Grammaticos, T. Tamizhmani, A. Ramani and K. Tamizhmani, Growth and integrability in discrete systems, J. Phys. A 34 (2001), 3811–3821.
V. Gromak, I. Laine and S. Shimomura, Painlevé Differential Equations in the Complex Plane, Studies in Mathematics 28 (2002).
R. Halburd and R. Korhonen, Existence of finite order meromorphic solutions as a detector of integrability in difference equations, submitted.
J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and K. Tohge, Complex difference equations of Malmquist type, Comp. Methods Func. Theory 1 (2001), 27–39.
J. Heittokangas, I. Laine, J. Rieppo and D. Yang, Meromorphic solutions of some linear functional equations, Aequationes Math. 60 (2000), 148–166.
G. Hiromi and M. Ozawa, On the existence of analytic mappings between two ultrahyperelliptic surfaces, Kodai Math. Sem. Report 17 (1965), 281–306.
I. Laine, Nevanlinna Theory and Complex Differential Equations, Studies in Mathematics 15, W. de Gruyter, Berlin, 1993.
A. Shidlovskii, Transcendental Numbers, Studies in Mathematics 12, W. de Gruyter, Berlin, 1989.
H. Silvennoinen, Meromorphic solutions of some composite functional equations, Ann. Acad. Sci. Fenn. Math. Diss. 133 (2003), 1–39.
G. Weissenborn, On the theorem of Tumura and Clunie, Bull. London Math. Soc. 18 (1986), 371–373.
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by the Academy of Finland grants 50981 and 210245, as well as by the Väisalä Fund of the Finnish Academy of Sciences and Letters.
Rights and permissions
About this article
Cite this article
Laine, I., Rieppo, J. & Silvennoinen, H. Remarks on Complex Difference Equations. Comput. Methods Funct. Theory 5, 77–88 (2005). https://doi.org/10.1007/BF03321087
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03321087