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Value Distribution and Linear Operators

Published online by Cambridge University Press:  22 November 2013

R. Halburd  and
R. Korhonen
R. Halburd
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK, ([email protected])
R. Korhonen
Affiliation:
Department of Physics and Mathematics, University of Eastern Finland, PO Box 111, 80101 Joensuu, Finland, ([email protected])
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Abstract

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Nevanlinna's second main theorem is a far-reaching generalization of Picard's theorem concerning the value distribution of an arbitrary meromorphic function f. The theorem takes the form of an inequality containing a ramification term in which the zeros and poles of the derivative f′ appear. We show that a similar result holds for special subfields of meromorphic functions where the derivative is replaced by a more general linear operator, such as higher-order differential operators and differential-difference operators. We subsequently derive generalizations of Picard's theorem and the defect relations.

Keywords

Nevanlinnadifferential differenceramification termPicard's theoremdefect relationslinear operatorPrimary 30D35Secondary 34M0334M0539A06
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 2 , June 2014 , pp. 493 - 504
Copyright
Copyright © Edinburgh Mathematical Society 2014 

References

1.Ablowitz, M. J., Halburd, R. and Herbst, B., On the extension of the Painlevé property to difference equations, Nonlinearity 13 (2000), 889–905.Google Scholar
2.Barnett, D., Halburd, R. G., Korhonen, R. J. and Morgan, W., Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equation, Proc. R. Soc. Edinb. A 137 (2007), 457–474.Google Scholar
3.Cherry, W. and Ye, Z., Nevanlinna's theory of value distribution (Springer, 2001).CrossRefGoogle Scholar
4.Chiang, Y. M. and Feng, S. J., On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane, Ramanujan J. 16 (2008), 105–129.CrossRefGoogle Scholar
5.Halburd, R. G. and Korhonen, R. J., Nevanlinna theory for the difference operator, Annales Acad. Sci. Fenn. Math. 31 (2006), 463–478.Google Scholar
6.Halburd, R. G. and Korhonen, R. J., Meromorphic solutions of difference equations, integrability and the discrete Painleve equations, J. Phys. A 40 (2007), R1–R38.Google Scholar
7.Halburd, R. G., Korhonen, R. and Tohge, K., Holomorphic curves with shift-invariant hyperplane preimages, Trans. Amer. Math. Soc. (in press).Google Scholar
8.Hayman, W. K., Meromorphic functions (Clarendon, Oxford, 1964).Google Scholar
9.Laine, I., Nevanlinna theory and complex differential equations (Walter de Gruyter, Berlin, 1993).Google Scholar
10.McMillan, E. M., A problem in the stability of periodic systems, in Topics in modern physics, a tribute to E. U. Condon (ed. Brittin, W. E. and Odabasi, H.), pp. 219–244 (Colorado Associated University Press, Boulder, CO, 1971).Google Scholar
11.Quispel, G. R. W., Capel, H. W. and Sahadevan, R., Continuous symmetries of differential-difference equations: the Kac–van Moerbeke equation and Painleve reduction, Phys. Lett. A 170 (1992), 379–383.Google Scholar