Some properties of meromorphic solutions of systems of complex \(q\)-shift difference equations. (English) Zbl 1272.30048

Xu, Hong-Yan; Liu, Bing-Xiang; Tang, Ke-Zong

Abstr. Appl. Anal. 2013, Article ID 680956, 6 p. (2013).

Summary: In view of Nevanlinna theory, we study the properties of meromorphic solutions of systems of a class of complex difference equations. Some results obtained improve and extend the previous theorems given by Gao.

MSC:

30D35	Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
39B32	Functional equations for complex functions
39A10	Additive difference equations

References:

[1]	Hayman, W. K., Meromorphic Functions. Meromorphic Functions, Oxford Mathematical Monographs (1964), Oxford, UK: Clarendon Press, Oxford, UK · Zbl 0115.06203
[2]	Yang, L., Value Distribution Theory (1993), Berlin, Germany: Springer, Berlin, Germany
[3]	Yang, C.-C.; Yi, H.-X., Uniqueness Theory of Meromorphic Functions. Uniqueness Theory of Meromorphic Functions, Mathematics and its Applications, 557 (2003), Dordrecht, The Netherlands: Kluwer Academic, Dordrecht, The Netherlands · Zbl 1070.30011
[4]	Ablowitz, M. J.; Halburd, R.; Herbst, B., On the extension of the Painlevé property to difference equations, Nonlinearity, 13, 3, 889-905 (2000) · Zbl 0956.39003
[5]	Barnett, D. C.; Halburd, R. G.; Morgan, W.; Korhonen, R. J., Nevanlinna theory for the \(q\)-difference operator and meromorphic solutions of \(q\)-difference equations, Proceedings of the Royal Society of Edinburgh A, 137, 3, 457-474 (2007) · Zbl 1137.30009
[6]	Gundersen, G. G.; Heittokangas, J.; Laine, I.; Rieppo, J.; Yang, D., Meromorphic solutions of generalized Schröder equations, Aequationes Mathematicae, 63, 1-2, 110-135 (2002) · Zbl 1012.30016
[7]	Heittokangas, J.; Korhonen, R.; Laine, I.; Rieppo, J.; Tohge, K., Complex difference equations of Malmquist type, Computational Methods and Function Theory, 1, 1, 27-39 (2001) · Zbl 1013.39001
[8]	Heittokangas, J.; Korhonen, R.; Laine, I.; Rieppo, J.; Zhang, J., Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity, Journal of Mathematical Analysis and Applications, 355, 1, 352-363 (2009) · Zbl 1180.30039
[9]	Zhang, J.; Korhonen, R., On the Nevanlinna characteristic of \(f(q z)\) and its applications, Journal of Mathematical Analysis and Applications, 369, 2, 537-544 (2010) · Zbl 1198.30033
[10]	Chiang, Y.-M.; Feng, S.-J., On the Nevanlinna characteristic of \(f(z + \eta)\) and difference equations in the complex plane, Ramanujan Journal, 16, 1, 105-129 (2008) · Zbl 1152.30024
[11]	Halburd, R. G.; Korhonen, R. J., Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, Journal of Mathematical Analysis and Applications, 314, 2, 477-487 (2006) · Zbl 1085.30026
[12]	Bergweiler, W.; Ishizaki, K.; Yanagihara, N., Growth of meromorphic solutions of some functional equations. I, Aequationes Mathematicae, 63, 1-2, 140-151 (2002) · Zbl 1009.39022
[13]	Chen, Z.-X., Growth and zeros of meromorphic solution of some linear difference equations, Journal of Mathematical Analysis and Applications, 373, 1, 235-241 (2011) · Zbl 1208.39028
[14]	Halburd, R. G.; Korhonen, R. J., Finite-order meromorphic solutions and the discrete Painlevé equations, Proceedings of the London Mathematical Society, 94, 2, 443-474 (2007) · Zbl 1119.39014
[15]	Halburd, R. G.; Korhonen, R. J., Nevanlinna theory for the difference operator, Annales Academiæ Scientiarum Fennicæ, 31, 2, 463-478 (2006) · Zbl 1108.30022
[16]	Laine, I.; Rieppo, J.; Silvennoinen, H., Remarks on complex difference equations, Computational Methods and Function Theory, 5, 1, 77-88 (2005) · Zbl 1093.39018
[17]	Silvennoinen, H., Meromorphic Solutions of Some Composite Functional Equations. Meromorphic Solutions of Some Composite Functional Equations, Annales Academiae Scientiarum Fennicae, Mathematica, Dissertationes, 132 (2003), Helsinki · Zbl 1026.30026
[18]	Wang, J., Growth and poles of meromorphic solutions of some difference equations, Journal of Mathematical Analysis and Applications, 379, 1, 367-377 (2011) · Zbl 1222.30023
[19]	Zheng, X.-M.; Chen, Z.-X., Some properties of meromorphic solutions of \(q\)-difference equations, Journal of Mathematical Analysis and Applications, 361, 2, 472-480 (2010) · Zbl 1185.39006
[20]	Korhonen, R., A new Clunie type theorem for difference polynomials, Journal of Difference Equations and Applications, 17, 3, 387-400 (2011) · Zbl 1213.39005
[21]	Zheng, X. M.; Chen, Z. X., Value distribution of meromorphic solutions of some \(q\)-difference equations, Journal of Mathematical Research and Exposition, 31, 4, 698-704 (2011) · Zbl 1249.30109
[22]	Gao, L., On meromorphic solutions of a type of system of composite functional equations, Acta Mathematica Scientia B, 32, 2, 800-806 (2012) · Zbl 1265.39037
[23]	Gao, L. Y., Systems of complex difference equations of Malmquist type, Acta Mathematica Sinica, 55, 2, 293-300 (2012) · Zbl 1265.39035
[24]	Gao, L., Estimates of \(N\)-function and \(m\)-function of meromorphic solutions of systems of complex difference equations, Acta Mathematica Scientia B, 32, 4, 1495-1502 (2012) · Zbl 1274.39037
[25]	Wang, H.; Xu, H. Y.; Liu, B. X., The poles and growth of solutions of systems of complex difference equations, Advances in Difference Equations, 2013 (2013)
[26]	Xu, H.-Y.; Cao, T.-B.; Liu, B.-X., The growth of solutions of systems of complex q-shift difference equations, Advances in Difference Equations, 2012 (2012) · Zbl 1377.39016
[27]	Liu, K.; Qi, X.-G., Meromorphic solutions of \(q\)-shift difference equations, Annales Polonici Mathematici, 101, 3, 215-225 (2011) · Zbl 1235.30023
[28]	Laine, I., Nevanlinna Theory and Complex Differential Equations. Nevanlinna Theory and Complex Differential Equations, de Gruyter Studies in Mathematics, 15 (1993), Berlin, Germany: Walter de Gruyter, Berlin, Germany · Zbl 0784.30002
[29]	Xu, H. Y.; Tu, J., Some properties of meromorphic solutions of \(q\)-shift difference equations, Journal of Mathematical Study, 45, 2, 124-132 (2012)