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摘要: 本文主要研究了非线性差分方程亚纯函数解的增长性问题。利用Nevanlinna理论重点研究了fn+P(z)(Δcf)m=Q(z)非線性差分方程的整函数解的一些基本性质，给出了在某些条件下其亚纯函数解具有一些特殊的形式。证明中主要应用了Hadamard分解来给出亚纯函数解的形式，同时也考虑了该方程的解的增长级问题，深化和改良了早期的一些结果。
关键词: 差分多项式；增长级；亚纯函数；Nevanlinna理论
【中图分类号】O174【文献标识码】B
【文章编号】2236-1879(2017)08-0260-02
参考文献
［1］W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
［2］I. Laine, Nevanlinna Theory and Complex Differential Equations,W.de.Gruyter, Berlin, 1993.
［3］H. X. Yi, C. C. Yang, The Uniqueness Theory of Meromorphic Functions, Science Press, Beijing, 1995.
［4］L. Yang, Value Distribution Theory and New Research, Science Press, Beijing, 1982.
［5］J. M. Whittaker, Interpolatory Function Theory, Cambridge Tracts in Math. and Math. Phys., vol 33, Cambridge University Press, 1935.
［6］F. Gross, On the equationfn+gn=1, Bull. Amer. Math. Soc. 72(1966),86-88.
［7］K. Liu, Meromorphic functions sharing a set with applications to difference equations, J. Math. Anal. Appl. 359(2009),384-393.
［8］X. G. Qi, Value distribution and uniqueness of difference polynomials and entire solutions of difference equations, unpublished information.
［9］Chen Z. X., On growth, zeros and poles of meromorphic solutions of linear and nonlinear difference equations, Sci. China Math, 54(2011), 2123-2133.
［10］R. G. Halburd and R. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Anal. Math. Appl. 314(2006), 477-487.
［11］I. Laine and C. C. Yang, Clunie theorems for difference and q-difference polynomial, J. London Math Soc., 83(2007), 148-152.
［12］S. Bank, A general theorem concerning the growth of solutions of first-order algebraic differential equations, Compositio Math., 25(1972), 61-70.
［13］G. Gundersen, Finite order solutions of second order linear differential equations, Trans. Amer. Math. Soc., 305(1988), 415-429.
关键词: 差分多项式；增长级；亚纯函数；Nevanlinna理论
【中图分类号】O174【文献标识码】B
【文章编号】2236-1879(2017)08-0260-02
参考文献
［1］W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
［2］I. Laine, Nevanlinna Theory and Complex Differential Equations,W.de.Gruyter, Berlin, 1993.
［3］H. X. Yi, C. C. Yang, The Uniqueness Theory of Meromorphic Functions, Science Press, Beijing, 1995.
［4］L. Yang, Value Distribution Theory and New Research, Science Press, Beijing, 1982.
［5］J. M. Whittaker, Interpolatory Function Theory, Cambridge Tracts in Math. and Math. Phys., vol 33, Cambridge University Press, 1935.
［6］F. Gross, On the equationfn+gn=1, Bull. Amer. Math. Soc. 72(1966),86-88.
［7］K. Liu, Meromorphic functions sharing a set with applications to difference equations, J. Math. Anal. Appl. 359(2009),384-393.
［8］X. G. Qi, Value distribution and uniqueness of difference polynomials and entire solutions of difference equations, unpublished information.
［9］Chen Z. X., On growth, zeros and poles of meromorphic solutions of linear and nonlinear difference equations, Sci. China Math, 54(2011), 2123-2133.
［10］R. G. Halburd and R. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Anal. Math. Appl. 314(2006), 477-487.
［11］I. Laine and C. C. Yang, Clunie theorems for difference and q-difference polynomial, J. London Math Soc., 83(2007), 148-152.
［12］S. Bank, A general theorem concerning the growth of solutions of first-order algebraic differential equations, Compositio Math., 25(1972), 61-70.
［13］G. Gundersen, Finite order solutions of second order linear differential equations, Trans. Amer. Math. Soc., 305(1988), 415-429.