대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제55권6호
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  • Pages.1845-1858
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  • 2018
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  • 1015-8634(pISSN)
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  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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THE GENERALIZED FERMAT TYPE DIFFERENCE EQUATIONS

  • Liu, Kai (Department of Mathematics Nanchang University) ;
  • Ma, Lei (Department of Mathematics Nanchang University) ;
  • Zhai, Xiaoyang (Dongfang College Shandong University of Finance and Economics)
  • 투고 : 2017.12.24
  • 심사 : 2018.04.03
  • 발행 : 2018.11.30
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초록

This paper is to consider the generalized Fermat difference equations with different types which ever considered by Li [14], Ishizaki and Korhonen [9], Zhang [26] and Liu [15-18], respectively. Some new observations and results on these equations will be given.

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참고문헌

  1. I. N. Baker, On a class of meromorphic functions, Proc. Amer. Math. Soc. 17 (1966), 819-822. https://doi.org/10.1090/S0002-9939-1966-0197732-X
  2. Y. M. Chiang and S. J. Feng, On the Nevanlinna characteristic f(z + ${\eta}$) and difference equations in the complex plane, Ramanujan. J. 16 (2008), 105-129. https://doi.org/10.1007/s11139-007-9101-1
  3. F. Gross, On the equation $f^n$ + $g^n$ = 1, Bull. Amer. Math. Soc. 72 (1966), 86-88. https://doi.org/10.1090/S0002-9904-1966-11429-5
  4. F. Gross, On the functional equation $f^n$ + $g^n$ = $h^n$, Amer. Math. Monthly 73 (1966), 1093-1096. https://doi.org/10.2307/2314644
  5. G. G. Gundersen, Complex functional equations, in Complex differential and functional equations (Mekrijarvi, 2000), 21-50, Univ. Joensuu Dept. Math. Rep. Ser., 5, Univ. Joensuu, Joensuu, 2003.
  6. G. G. Gundersen and W. K. Hayman, The strength of Cartan's version of Nevanlinna theory, Bull. London Math. Soc. 36 (2004), no. 4, 433-454. https://doi.org/10.1112/S0024609304003418
  7. R. G. Halburd and R. J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 2, 463-478.
  8. R. Halburd, R. Korhonen, and K. Tohge, Holomorphic curves with shift-invariant hyperplane preimages, Trans. Amer. Math. Soc. 366 (2014), no. 8, 4267-4298. https://doi.org/10.1090/S0002-9947-2014-05949-7
  9. K. Ishizaki and R. Korhonen, Meromorphic solutions of algebraic difference equations, Constr Approx (2017); https://doi.org/10.1007/s00365-017-9401-7.
  10. D.-I. Kim, Waring's problem for linear polynomials and Laurent polynomials, Rocky Mountain J. Math. 35 (2005), no. 5, 1533-1553. https://doi.org/10.1216/rmjm/1181069650
  11. R. Korhonen and Y. Y. Zhang, Existence of meromorphic solutions of first order difference equations, arXiv:1708.07647.
  12. I. Lahiri and K.-W. Yu, On generalized Fermat type functional equations, Comput. Methods Funct. Theory 7 (2007), no. 1, 141-149. https://doi.org/10.1007/BF03321637
  13. N. Li, On the existence of solutions of a Fermat-type difference equation, Ann. Acad. Sci. Fenn. Math. 41 (2016), no. 2, 523-549. https://doi.org/10.5186/aasfm.2016.4131
  14. P. Li and C.-C. Yang, Some further results on the unique range sets of meromorphic functions, Kodai Math. J. 18 (1995), no. 3, 437-450. https://doi.org/10.2996/kmj/1138043482
  15. K. Liu, Meromorphic functions sharing a set with applications to difference equations, J. Math. Anal. Appl. 359 (2009), no. 1, 384-393. https://doi.org/10.1016/j.jmaa.2009.05.061
  16. K. Liu, T. Cao, and H. Cao, Entire solutions of Fermat type differential-difference equations, Arch. Math. (Basel) 99 (2012), no. 2, 147-155. https://doi.org/10.1007/s00013-012-0408-9
  17. K. Liu and L. Yang, On entire solutions of some differential-difference equations, Comput. Methods Funct. Theory 13 (2013), no. 3, 433-447. https://doi.org/10.1007/s40315-013-0030-2
  18. K. Liu and L. Z. Yang, A note on meromorphic solutions of Fermat types equations, An. Stiint. Univ. Al. I. Cuza Iasi Mat. (N.S.) 1 (2016), 317-325.
  19. F. Lu and Q. Han, On the Fermat-type equation $f^3(z)$ + $f^3(z+c)$ = 1, Aequationes Math. 91 (2017), no. 1, 129-136. https://doi.org/10.1007/s00010-016-0443-x
  20. P. Montel, Lecons sur les familles normales de fonctions analytiques at leurs applications, Gauthier-Villars, Paris, (1927), 135-136 (French).
  21. T. W. Ng and S. K. Yeung, Entire holomorphic curves on a Fermat surface of low degree, arXiv:1612.01290 (2016).
  22. N. Toda, On the functional equation ${\Sigma}_{i=0}^pa_if^{n_i}_i$ = 1, Tohoku Math. J. (2) 23 (1971), 289-299. https://doi.org/10.2748/tmj/1178242646
  23. C.-C. Yang and P. Li, On the transcendental solutions of a certain type of nonlinear differential equations, Arch. Math. (Basel) 82 (2004), no. 5, 442-448. https://doi.org/10.1007/s00013-003-4796-8
  24. C.-C. Yang and H.-X. Yi, Uniqueness Theory of Meromorphic Functions, Mathematics and its Applications, 557, Kluwer Academic Publishers Group, Dordrecht, 2003.
  25. K.-W. Yu and C.-C. Yang, A note for Waring's type of equations for the ring of meromorphic functions, Indian J. Pure Appl. Math. 33 (2002), no. 10, 1495-1502.
  26. J. Zhang, On some special difference equations of Malmquist type, Bull. Korean. Math. Soc. 55 (2018), no. 1, 51-61. https://doi.org/10.4134/BKMS.B160844