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Complex Difference Equations of Malmquist Type

Complex Difference Equations of Malmquist Type 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Complex Difference Equations of Malmquist Type

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Publisher
Springer Journals
Copyright
Copyright © 2001 by Heldermann Verlag
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/BF03320974
Publisher site
See Article on Publisher Site

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.

Journal

Computational Methods and Function Theory – Springer Journals

Published: Mar 7, 2013

References

  • On the extension of the Painlevé property to difference equations
    Ablowitz, M J; Halburd, R; Herbst, B
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    Bank, S; Kaufman, R
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    Carleson, L; Gamelin, T
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    Clunie, J
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    Gundersen, G; Heittokangas, J; Laine, I; Rieppo, J; Yang, D
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    Hayman, W K
  • Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen
    Jank, G; Volkmann, L
  • Nevanlinna Theory and Complex Differential Equations
    Laine, I
  • Entire solutions of a polynomial difference equation
    Shimomura, S
  • Meromorphic solutions of some difference equations
    Yanagihara, N