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Value distribution of meromorphic solutions of certain difference Painlevé equations. (English) Zbl 1183.30026
Summary: The Borel exceptional value and the exponents of convergence of poles, zeros, and fixed points of finite order transcendental meromorphic solutions for difference Painlevé \(I\) and \(II\) equations are estimated. The forms of rational solutions of the difference Painlevé \(II\) equation and the autonomous difference Painlevé \(I\) equation are also given. It is also proved that the non-autonomous difference Painlevé \(I\) equation has no rational solution.

MSC:
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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[1] Ablowitz, M.; Halburd, R.G.; Herbst, B., On the extension of Painlevé property to difference equations, Nonlinearity, 13, 889-905, (2000) · Zbl 0956.39003
[2] Bergweiler, W.; Langley, J.K., Zeros of differences of meromorphic functions, Math. proc. Cambridge philos. soc., 142, 133-147, (2007) · Zbl 1114.30028
[3] Chen, Z.X.; Shon, K.H., On zeros and fixed points of differences of meromorphic functions, J. math. anal. appl., 344, 373-383, (2008) · Zbl 1144.30012
[4] Chiang, Y.M.; Feng, S.J., On the Nevanlinna characteristic of \(f(z + \eta)\) and difference equations in the complex plane, Ramanujan J., 16, 105-129, (2008) · Zbl 1152.30024
[5] Fokas, A.S., From continuous to discrete Painlevé equations, J. math. anal. appl., 180, 342-360, (1993) · Zbl 0794.34013
[6] Halburd, R.G.; Korhonen, R., Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. math. anal. appl., 314, 477-487, (2006) · Zbl 1085.30026
[7] Halburd, R.G.; Korhonen, R., Meromorphic solution of difference equation, integrability and the discrete Painlevé equations, J. phys. A, 40, 1-38, (2007) · Zbl 1115.39024
[8] Halburd, R.G.; Korhonen, R., Finite-order meromorphic solutions and the discrete Painlevé equations, Proc. lond. math. soc., 94, 443-474, (2007) · Zbl 1119.39014
[9] Halburd, R.G.; Korhonen, R., Existence of finite-order meromorphic solutions as a detector of integrability in difference equations, Phys. D, 218, 191-203, (2006) · Zbl 1105.39019
[10] Hayman, W.K., Meromorphic functions, (1964), Clarendon Press Oxford · Zbl 0115.06203
[11] Heittokangas, J.; Korhonen, R.; Laine, I.; Rieppo, J.; Hohge, K., Complex difference equations of Malmquist type, Comput. methods funct. theory, 1, 27-39, (2001) · Zbl 1013.39001
[12] Heittokangas, J.; Korhonen, R.; Laine, I.; Rieppo, J.; Zhang, J., Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity, J. math. anal. appl., 355, 352-363, (2009) · Zbl 1180.30039
[13] Laine, I.; Yang, C.C., Clunie theorems for difference and q-difference polynomials, J. lond. math. soc., 76, 3, 556-566, (2007) · Zbl 1132.30013
[14] Laine, I., Nevanlinna theory and complex differential equations, (1993), W. de Gruyter Berlin
[15] Painlevé, P., Mémoire sur LES équations différentielles dont l’integrale générale est uniforme, Bull. soc. math. France, 28, 201-261, (1900) · JFM 31.0337.03
[16] Shimomura, S., Entire solutions of a polynomial difference equation, J. fac. sci. univ. Tokyo sect. IA math., 28, 253-266, (1981) · Zbl 0469.30021
[17] Yanagihara, N., Meromorphic solutions of some difference equations, Funkcial. ekvac., 23, 309-326, (1980) · Zbl 0474.30024
[18] Yi, H.X.; Yang, C.C.; Yang, C.C.; Yi, H.X., Uniqueness theory of meromorphic functions, (2003), Kluwer Academic Publishers Group Dordrecht, (in Chinese); or · Zbl 0799.30019
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