Difference Nevanlinna theories with vanishing and infinite periods

By extending the idea of a difference operator with a fixed step to varying-steps difference operators, we have established a difference Nevanlinna theory for meromorphic functions with the steps tending to zero (vanishing period) and a difference Nevanlinna theory for finite order meromorphic functions with the steps tending to infinity (infinite period) in this paper. We can recover the classical little Picard theorem from the vanishing period theory, but we require additional finite order growth restriction for meromorphic functions from the infinite period theory. Then we give some applications of our theories to exhibit connections between discrete equations and and their continuous analogues.

[1]  K. Shon,et al.  Complex Differences and Difference Equations , 2014 .

[2]  A. Ramani,et al.  Discrete Painlevé equations: an integrability paradigm , 2014 .

[3]  R. Korhonen,et al.  Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations , 2007 .

[4]  Y. Chiang,et al.  On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions , 2006, math/0610480.

[5]  Y. Chiang,et al.  On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane , 2006, math/0609324.

[6]  Y. Chiang,et al.  On the Nevanlinna Order of Meromorphic Solutions to Linear Analytic Difference Equations , 2006 .

[7]  R. Korhonen,et al.  Nevanlinna theory for the difference operator , 2005, math/0506011.

[8]  R. Korhonen,et al.  Finite‐order meromorphic solutions and the discrete Painlevé equations , 2005, nlin/0504026.

[9]  R. Korhonen,et al.  Difference analogue of the Lemma on the Logarithmic Derivative with applications to difference equations , 2005, math/0504245.

[10]  B. Herbst,et al.  On the extension of the Painlevé property to difference equations , 2000 .

[11]  A. Ramani,et al.  Discrete Painlevé equations: coalescences, limits and degeneracies , 1995, solv-int/9510011.

[12]  Lo Yang,et al.  Value Distribution Theory , 1993 .

[13]  Ilpo Laine,et al.  Nevanlinna Theory and Complex Differential Equations , 1992 .

[14]  G. G. Gundersen,et al.  Estimates for the Logarithmic Derivative of a Meromorphic Function, Plus Similar Estimates , 1988 .

[15]  J. Miles Some Examples of the Dependence of the Nevanlinna Deficiency Upon the Choice of Origin , 1983 .