Abstract
In this paper, we mainly apply a new, asymptotic method to investigate the growth of meromorphic solutions of linear higher order difference equations and differential equations. We delete the condition (1.6) of Theorems E and F, yet obtain the same results for Theorems E and F. We also weaken the condition (1.4) of Theorems C and D.
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References
Bank S, Laine I, On the oscillation theory of f″ + Af = 0 where A is entire. Trans Amer Math Soc, 1982, 273: 351–363
Bergweiler W, Langley J K. Zeros of differences of meromorphic functions. Math Proc Camb Phil Soc, 2007, 142: 133–147
Chen C X, Chen Z X. Some results concerning meromorphic solutions for the Pielou Logistic equation. Bull Malays Math Sci Soc, 2020, 43: 1775–1792
Chen Z X. Complex Differences and Difference Equations, Mathematics Monograph Series 29. Beijing: Science Press, 2014
Chen Z X. Growth and zeros of meromorphic solution of some linear difference equations. J Math Anal Appl, 2011, 373: 235–241
Chen Z X. On growth, zeros and poles of meromorphic solutions of linear and nonlinear difference equations. Science China Math, 2011, 54(10): 2123–2133
Chen Z X. The growth of solutions of f″ + e−zf′ + Q(z)f = 0 where the order (Q) = 1. Science in China (series A), 2002, 45(3): 290–300
Chen Z X, Shon K H. On existence of solutions of difference Riccati equation. Acta Mathematica Scientia, 2019, 39B(1): 1–9
Chiang Y M, Gao S A. On a problem in complex oscillation theory of periodic second order linear differential equations and some related perturbation results. Ann Acad Sci Fenn Math, 2002, 27: 273–290
Chiang Y M, Feng S J. On the Nevanlinna characteristic of f (z + η) and difference equations in the complex plane. Ramanujan J, 2008, 16: 105–129
Cui N, Chen Z X. The conjecture on unity of meromorphic functions concerning their differences. J Difference Equa Appl, 2016, 22(10): 1452–1471
Frank G, Hellerstein S. On the meromorphic solutions of non-homogeneous linear differetial equations with polynomial coefficients. Proc London Math Soc, 1986, 53(3): 407–428
Gao S A, Chen Z X, Chen T W. Complex Oscillation Theory of Linear Differential Equations. Wuhan: Huazhong University of Science and Technology Press, 1998 (Chinese)
Gundersen G. Finite order solutions of second order linear differential equations. Trans Amer Math Soc, 1988, 305: 415–429
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
Hayman W K. Meromorphic Functions. Oxford: Clarendon Press, 1964
He Y Z, Xiao X Z. Algebroid Functions and Ordinary Differential Equations. Beijing: Science Press, 1988 (Chinese)
Ishizaki K, Wen Z T. Binomial series and complex difference equations. J Math Anal Appl, 2021, 497(1): Article 124844
Ishizaki K, Yanagihara N. Wiman-Valiron method for difference equations. Nagoya Math J, 2004, 175: 75–102
Laine I. Nevanlinna Theory and Complex Differential Equations. Berlin: W de Gruyter, 1993
Lan S T, Chen Z X. On fixed points of meromorphic functions f(z) and f(z+c), ∆cf(z). Acta Mathematica Scientia, 2019, 39B(5): 1277–1289
Yang L. Value Distribution Theory. Beijing: Science Press, 1993
Zhang R R, Chen Z X. Fixed points of meromorphic functions and of their differences, divided differences and shifts. Acta Mathematica Sinica, English Series, 2016, 32(10): 1189–1202
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Dedicated to the memory of Professor Jiarong YU
This work is supported by the National Natural Science Foundation of China (11771090, 11871260, 11761035, 11801093, 11801110).
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Chen, Z., Zhang, R., Lan, S. et al. The Growth of Difference Equations and Differential Equations. Acta Math Sci 41, 1911–1920 (2021). https://doi.org/10.1007/s10473-021-0608-0
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DOI: https://doi.org/10.1007/s10473-021-0608-0