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)
Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations. (English) Zbl 1115.39024

In this comprehensive survey article the authors firstly describe, by way of examples, some of the remarkable properties of discrete Painlevé equations, such as: the existence of a related linear (iso-monodromy) problem, the existence of Bäcklund transformations and relations to Bäcklund transformations of differential Painlevé equations. Then they present several theorems on the existence of non-trivial meromorphic solutions of certain classes of difference equations: linear difference equations, nonlinear first-order difference equations, the QRT map and some nonlinear higher order difference equations.

An introduction to basic Nevanlinna theory, with some properties of the characteristic function of a meromorphic function and the lemma on the logarithmic derivative are also given. This theory is applied to difference equations admitting finite-order meromorphic solutions. The authors present several strong necessary conditions for an equation to admit a meromorphic solution, by using some different tools: difference analogues of the lemma of the logarithmic derivative, J. Clunie’s lemma [Lond. Math. Soc. 37, 17–27 (1962; Zbl 0104.29504)] and a value distribution result of A. Z. Mokhon’ko and V. D. Mokhon’ko [Sib. Mat. Ab. 15, 1305–1322 (1974; Zbl 0303.30024)]. A new example of an equation of the form w(z+1)w(z-1)=R(z,w(z)), where R is rational in w with meromorphic coefficients, is studied.

An overview of recent results on meromorphic solutions of linear difference equations, such as: a version of Wiman-Valiron theory for slow growing functions [cf. W. K. Hayman, Can. Math. Bull. 17, 317–358 (1974; Zbl 0314.30021)], order estimates for the growth of finite-order solutions and a theorem concerning minimal solutions of first-order equations, is presented. The authors also present a number of recent results on the value distribution of shifts of finite-order meromorphic functions (difference analogues of the Nevanlinna’s second main theorem, Picard’s theorem, defect relations and theorems concernings meromorphic functions sharing values). They finally describe a q-difference analogue of the lemma on the logarithmic derivative, which leads to q-difference analogues of some theorems presented in the paper.

MSC:
39A13Difference equations, scaling (q-differences)
34M55Painlevé and other special equations; classification, hierarchies
30D35Distribution of values (one complex variable); Nevanlinna theory