Computational Methods and Function Theory

, Volume 1, Issue 1, pp 27–39

Complex Difference Equations of Malmquist Type

  • Janne Heittokangas
  • Risto Korhonen
  • Ilpo Laine
  • Jarkko Rieppo
  • Kazuya Tohge
Article

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 degyR(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.

Keywords

Complex difference equation value distribution Nevanlinna characteristic Borel exceptional values 

2000 MSC

39A10 30D35 39A12 

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Copyright information

© Heldermann Verlag 2001

Authors and Affiliations

  • Janne Heittokangas
    • 1
  • Risto Korhonen
    • 1
  • Ilpo Laine
    • 1
  • Jarkko Rieppo
    • 1
  • Kazuya Tohge
    • 2
  1. 1.Department of MathematicsUniversity of JoensuuJoensuuFinland
  2. 2.Faculty of TechnologyKanazawa UniversityKanazawaJapan

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