zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Remarks on complex difference equations. (English) Zbl 1093.39018
Authors’ abstract: {\it R. Halburd} and {\it R. Korhonen} [Existence of finite order meromorphic solutions as a detector of integrability in difference equations. (to appear)] 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 $\text{deg}_f R(z,f) \leq 2$ 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.

39A12Discrete version of topics in analysis
30D05Functional equations in the complex domain, iteration and composition of analytic functions
30D35Distribution of values (one complex variable); Nevanlinna theory
39B12Iterative and composite functional equations
39B32Functional equations for complex functions
34M55Painlevé and other special equations; classification, hierarchies
39A10Additive difference equations
Full Text: DOI
[1] M. 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] P. M. Cohn, Algebra, Vol. 1. John Wiley & Sons, London - New York - Sydney, (1974).
[3] R. Goldstein, Some results on factorisation of meromorphic functions, J. London Math. Soc. (2) 4 (1971), 357--364. · Zbl 0223.30036 · doi:10.1112/jlms/s2-4.2.357
[4] R. Goldstein, On meromorphic solutions of certain functional equations, Aequationes Math. 18 (1978), 112--157. · Zbl 0384.30009 · doi:10.1007/BF01844071
[5] B. Grammaticos, T. Tamizhmani, A. Ramani and K. Tamizhmani, Growth and integrability in discrete systems, J. Phys. A 34 (2001), 3811--3821. · Zbl 1006.39018 · doi:10.1088/0305-4470/34/18/309
[6] V. Gromak, I. Laine and S. Shimomura, Painlevé Differential Equations in the Complex Plane, Studies in Mathematics 28 (2002). · Zbl 1043.34100
[7] R. Halburd and R. Korhonen, Existence of finite order meromorphic solutions as a detector of integrability in difference equations, submitted. · Zbl 1105.39019
[8] 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. · Zbl 1013.39001 · doi:10.1007/BF03320974
[9] J. Heittokangas, I. Laine, J. Rieppo and D. Yang, Meromorphic solutions of some linear functional equations, Aequationes Math. 60 (2000), 148--166. · Zbl 0963.39028 · doi:10.1007/s000100050143
[10] G. Hiromi and M. Ozawa, On the existence of analytic mappings between two ultrahyperelliptic surfaces, Kodai Math. Sem. Report 17 (1965), 281--306. · Zbl 0154.07903 · doi:10.2996/kmj/1138845125
[11] I. Laine, Nevanlinna Theory and Complex Differential Equations, Studies in Mathematics 15, W. de Gruyter, Berlin, 1993. · Zbl 0784.30002
[12] A. Shidlovskii, Transcendental Numbers, Studies in Mathematics 12, W. de Gruyter, Berlin, 1989.
[13] H. Silvennoinen, Meromorphic solutions of some composite functional equations, Ann. Acad. Sci. Fenn. Math. Diss. 133 (2003), 1--39. · Zbl 1026.30026
[14] G. Weissenborn, On the theorem of Tumura and Clunie, Bull. London Math. Soc. 18 (1986), 371--373. · Zbl 0598.30044 · doi:10.1112/blms/18.4.371