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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.
30D35Distribution of values (one complex variable); Nevanlinna theory
[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 · doi:10.1088/0951-7715/13/3/321
[2]Bergweiler, W.; Langley, J. K.: Zeros of differences of meromorphic functions, Math. proc. Cambridge philos. Soc. 142, 133-147 (2007) · Zbl 1114.30028 · doi:10.1017/S0305004106009777
[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 · doi:10.1016/j.jmaa.2008.02.048
[4]Chiang, Y. M.; Feng, S. J.: On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane, Ramanujan J. 16, 105-129 (2008) · Zbl 1152.30024 · doi:10.1007/s11139-007-9101-1
[5]Fokas, A. S.: From continuous to discrete Painlevé equations, J. math. Anal. appl. 180, 342-360 (1993)
[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 · doi:10.1016/j.jmaa.2005.04.010
[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 · doi:10.1088/1751-8113/40/6/R01
[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 · doi:10.1112/plms/pdl012
[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 · doi:10.1016/j.physd.2006.05.005
[10]Hayman, W. K.: Meromorphic functions, (1964) · 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 · doi:http://www.heldermann.de/CMF/CMF01/cmf0102.htm
[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 · doi:10.1016/j.jmaa.2009.01.053
[13]Laine, I.; Yang, C. C.: Clunie theorems for difference and q-difference polynomials, J. lond. Math. soc. 76, No. 3, 556-566 (2007) · Zbl 1132.30013 · doi:10.1112/jlms/jdm073
[14]Laine, I.: Nevanlinna theory and complex differential equations, (1993)
[15]Painlevé, P.: Mémoire sur LES équations différentielles dont l’integrale générale est uniforme, Bull. soc. Math. France 28, 201-261 (1900) · Zbl 31.0337.03 · doi:numdam:BSMF_1900__28__201_0
[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.: The uniqueness theory of meromorphic functions, (1995)