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On growth, zeros and poles of meromorphic solutions of linear and nonlinear difference equations. (English) Zbl 1238.30019
The author studies the order of growth, zeros and poles of finite order meromorphic solutions of nonlinear difference equations $$ P(z) y(z+1)y(z)+Q(z)y(z)y(z-1)=H(z)\eqno{(1.1)} $$ and $$ y(z+1)=\frac{R(z)y(z)}{Q(z)+P(z)y(z)},\eqno{(1.2)} $$ where $P(z), Q(z), H(z)$ and $R(z)$ are polynomials such that $P(z) Q(z) H(z) R(z)\not\equiv 0$. Similar results for linear difference equations related to (1.1) and (1.2) are also obtained.

30D35Distribution of values (one complex variable); Nevanlinna theory
39A10Additive difference equations
Full Text: DOI
[1] Ablowitz M, Halburd R G, Herbst B. On the extension of Painlevé property to difference equations. Nonlinearty 2000, 13: 889--905 · Zbl 0956.39003 · doi:10.1088/0951-7715/13/3/321
[2] Bank S. A general theorem concerning the growth of solutions of first-order algebraic differentiial equtions. Compositio Math, 1972, 25: 61--70 · Zbl 0246.34006
[3] Bergweiler W, Langley J K. Zeros of differences of meromorphic functions. Math Proc Camb Phil Soc, 2007, 142: 133--147 · Zbl 1114.30028 · doi:10.1017/S0305004106009777
[4] Chen Z X. Growth and zeros of meromorphic solution of some linear difference equations. J Math Anal Appl, 2011, 373: 235--241 · Zbl 1208.39028 · doi:10.1016/j.jmaa.2010.06.049
[5] Chen Z X, Shon K H. On zeros and fixed points of differencers of meromorphic functions. J Math Anal Appl, 2008, 344: 373--383 · Zbl 1144.30012 · doi:10.1016/j.jmaa.2008.02.048
[6] Chen Z X, Shon K H. Estimates for zeros of differences of meromorphic functions. Sci China Ser A, 2009, 52: 2447--2458 · Zbl 1181.30016 · doi:10.1007/s11425-009-0159-7
[7] Chen Z X, Shon K H. Value distribution of meromorphic solutions of certain difference Painlevé equations. J Math Anal Appl, 2010, 364: 556--566 · Zbl 1183.30026 · doi:10.1016/j.jmaa.2009.10.021
[8] Chiang Y M, Feng S J. On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. Trans Amer Math Soc, 2009, 361: 3767--3791 · Zbl 1172.30009 · doi:10.1090/S0002-9947-09-04663-7
[9] Chiang Y M, Feng S J. On the Nevanlinna characteristic of f(z + {$\nu$}) and difference equations in the complex plane. Ramanujan J, 2008, 16: 105--129 · Zbl 1152.30024 · doi:10.1007/s11139-007-9101-1
[10] Chiang Y M, Ruijsenaars S N M. On the Nevanlinna order of meromorphic solutions to linear analytic difference equations. Stud Appl Math, 2006, 116: 257--287 · Zbl 1145.39300 · doi:10.1111/j.1467-9590.2006.00343.x
[11] Gao S A, Chen Z X, Chen T W. Complex Oscillation Theory of Linear Differential Equations (in Chinese). Wuhan: Huazhong University of Science and Technology Press, 1998
[12] Gundersen G. Finite order solutions of second order linear differential equations. Trans Amer Math Soc, 1988, 305: 415--429 · Zbl 0634.34004 · doi:10.1090/S0002-9947-1988-0920167-5
[13] Gross F. Factorization of Meromorhpic Functions. Washinton, D. C.: Government Printing Office, 1972
[14] Halburd R G, Korhonen R. Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J Math Anal Appl, 2006, 314: 477--487 · Zbl 1085.30026 · doi:10.1016/j.jmaa.2005.04.010
[15] Halburd R G, Korhonen R. Existence of finite-order meromorphic solutions as a detector of integrability in difference equations. Physics D, 2006, 218: 191--203 · Zbl 1105.39019 · doi:10.1016/j.physd.2006.05.005
[16] Halburd R G, Korhonen R. Meromorphic solution of difference equation, integrability and the discrete Painlevé equations. J Phys A: Math Theor, 2007, 40: 1--38 · Zbl 1104.82019 · doi:10.1088/1751-8113/40/1/001
[17] Halburd R G, Korhonen R. Finite-order meromorphic solutions and the discrete Painlevé equations. Proc London Math Soc, 2007, 94: 443--474 · Zbl 1119.39014 · doi:10.1112/plms/pdl012
[18] Hayman W K. Meromorphic Functions. Oxford: Clarendon Press, 1964
[19] Hayman W K. The local growth of power series: a survey of the Wiman-Valiron method. Canad Math Bull, 1974, 17: 317--358 · Zbl 0314.30021 · doi:10.4153/CMB-1974-064-0
[20] Heittokangas J, Korhonen R, Laine I, et al. Complex difference equations of Malmquist type. Comput Methods Funct Theory, 2001, 1: 27--39 · Zbl 1013.39001 · doi:10.1007/BF03320974
[21] Heittokangas J, Korhonen R, Laine I, et al. Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity. J Math Anal Appl, 2009, 355: 352--363 · Zbl 1180.30039 · doi:10.1016/j.jmaa.2009.01.053
[22] Ishizaki K, Yanagihara N. Wiman-Valiron method for difference equations. Nagoya Math J, 2004, 175: 75--102 · Zbl 1070.39002
[23] Laine I, Yang C C. Clunie theorems for difference and q-difference polynomials. J London Math Soc, 2007, 76: 556--566 · Zbl 1132.30013 · doi:10.1112/jlms/jdm073
[24] Laine I, Yang C C. Value distribution of difference polynomials. Proc Japan Acad, 2007, 83: 148--151 · Zbl 1153.30030 · doi:10.3792/pjaa.83.148
[25] Laine I. Nevanlinna Theory and Complex Differential Equations. Berlin: Walter de Gruyter, 1993 · Zbl 0784.30002
[26] Yang C C, Yi H X. Uniqueness Theory of Meromorphic Functions. Dordrecht: Kluwer Academic Publishers Group, 2003 · Zbl 1070.30011
[27] Yang L. Value Distribution Theory. Beijing: Science Press, 1993 · Zbl 0790.30018