The Order of Growth of Meromorphic Solutions for Some Difference Painleve Equations
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摘要: 利用亚纯函数的Nevanlinna值分布理论的差分模拟，研究了给定的差分Painlevˊe方程I和差分Painlevˊe方程II的超越亚纯解的增长性，得到了一些有意义的结果：在给定的条件下，给出了给定的差分Painlevˊe方程I和差分Painlevˊe方程II的超越亚纯解的增长级的精确估计.
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关键词:
- 极点收敛指数
Abstract: By utilizing the difference analogue of Nevanlinna's value distribution theory of meromorphic functions, the order of growth of meromorphic solutions of certain difference Painlevˊe equation I and difference Painlevˊe equation II is investigated and some important results are obtained. The accurate estimate of the order of growth of meromorphic solutions to certain difference Painlevˊe equation I and difference Painlevˊe equation II is attained under the given conditions.-
Keywords:
- the exponent of convergence of poles
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[1] HAYMAN W K. Meromorphic functions[M]. Oxford: Clarendon Press, 1964.
[2] ABLOWITZ M, HALBURD R G, HERBST B. On the extention of Painlev′e property to difference equations[J]. Nonlinearity, 2000,13(1): 889–905.
[3] HALBURD R G, KORHONEN R J. Existence of finite-order meromorphic solutions as a detector of integrability of difference equations[J]. J. Phys., 2006, D(218): 191–203.
[4] HALBURD R G, KORHONEN R J. Meromorphic solution of difference equation, integrability and the discrete Painlev′e equations[J]. J. Phys., 2007, A(40): 1–38.
[5] HALBURD R G, KORHONEN R J. Finite-order meromorphic solutions and the discrete Painlev′e equations[J]. Proc. Lond. Math. Soc., 2007, 94(6): 443–474.
[6] CHEN Z X, SHON K H. Value distribution of meromorphic solutions of certain difference Painlev′e equations[J]. J. Math. Anal. Appl., 2010, 364(1): 556–566.
[7] CHEN M R, CHEN Z X. On properties of meromorphic solution of certain difference Painlev′e equation[J]. Bull. Aust. Math. Soc., 2012, 85(3): 463–475.
[8] CHEN Z X. On growth, zeros and poles of meromorphic solutions of linear and nonlinear difference equations[J]. Sci. China Math., 2011, 54(8): 2123–2133.
[9] PENG C W, CHEN Z X. On a conjecture concerning some nonlinear difference equations [J]. Bull. Malays. Math. Sci. Soc., 2013, 36 (2): 221–227.
[10]陈宗煊，黄志波.复域差分和差分方程的研究[J].华南师范大学学报(自然科学版), 2013, 45 (6): 26-33.
[11]蒋业阳，陈宗煊.某些差分方程的值分布[J]. 华南师范大学学报(自然科学版), 2013, 45 (1): 19-23.
[12]CHIANG Y M, FENG S J. On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane[J]. Ramanujan J., 2008, 16(1): 105-129.[1] HAYMAN W K. Meromorphic functions[M]. Oxford: Clarendon Press, 1964.
[2] ABLOWITZ M, HALBURD R G, HERBST B. On the extention of Painlev′e property to difference equations[J]. Nonlinearity, 2000,13(1): 889–905.
[3] HALBURD R G, KORHONEN R J. Existence of finite-order meromorphic solutions as a detector of integrability of difference equations[J]. J. Phys., 2006, D(218): 191–203.
[4] HALBURD R G, KORHONEN R J. Meromorphic solution of difference equation, integrability and the discrete Painlev′e equations[J]. J. Phys., 2007, A(40): 1–38.
[5] HALBURD R G, KORHONEN R J. Finite-order meromorphic solutions and the discrete Painlev′e equations[J]. Proc. Lond. Math. Soc., 2007, 94(6): 443–474.
[6] CHEN Z X, SHON K H. Value distribution of meromorphic solutions of certain difference Painlev′e equations[J]. J. Math. Anal. Appl., 2010, 364(1): 556–566.
[7] CHEN M R, CHEN Z X. On properties of meromorphic solution of certain difference Painlev′e equation[J]. Bull. Aust. Math. Soc., 2012, 85(3): 463–475.
[8] CHEN Z X. On growth, zeros and poles of meromorphic solutions of linear and nonlinear difference equations[J]. Sci. China Math., 2011, 54(8): 2123–2133.
[9] PENG C W, CHEN Z X. On a conjecture concerning some nonlinear difference equations [J]. Bull. Malays. Math. Sci. Soc., 2013, 36 (2): 221–227.
[10]陈宗煊，黄志波.复域差分和差分方程的研究[J].华南师范大学学报(自然科学版), 2013, 45 (6): 26-33.
[11]蒋业阳，陈宗煊.某些差分方程的值分布[J]. 华南师范大学学报(自然科学版), 2013, 45 (1): 19-23.
[12]CHIANG Y M, FENG S J. On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane[J]. Ramanujan J., 2008, 16(1): 105-129.
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