Home Mathematics Meromorphic solutions of q-shift difference equations
Article
Licensed
Unlicensed Requires Authentication

Meromorphic solutions of q-shift difference equations

  • Xiaoguang Qi EMAIL logo , Lianzhong Yang and Yong Liu
Published/Copyright: August 23, 2016
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 3

Abstract

In this paper, we consider the growth of meromorphic solutions of some generalized q-shift difference equation An(z)f(qz + n) + … + A1(z)f(qz + 1) + A0(z)f(qz) = 0. Moreover, we also investigate the existence of transcendental meromorphic solutions of Fermat type of q-shift difference equations.

MSC 2010: Primary 39A05; Secondary 30D35
Keywords: meromorphic functions; growth; q-shift difference equation

This work was supported by the National Natural Science Foundation of China (No. 11301220 and No. 11371225) and the Tianyuan Fund for Mathematics (No. 11226094), the NSF of Shandong Province, China (No. ZR2012AQ020 and No. ZR2010AM030) and the Fund of Doctoral Program Research of University of Jinan (XBS1211)

(Communicated by Stanisława Kanas)

Acknowledgement

The authors would like to thank the referee for his/her helpful suggestions and comments.

References

[1] Batchelder, P. M.: An Introduction to Linear Difference Equations, Dover Publications, Inc., New York, 1967.Search in Google Scholar

[2] Bergweiler, W.—Ishizaki, K.—Yanagihara, N.: Meromorphic solutions of some functional equations, Methods Appl. Anal. 6 (1999), 617–618. Correction: Methods Appl. Anal. 6 (1999), 617–618.10.4310/MAA.1998.v5.n3.a2Search in Google Scholar

[3] Chiang, Y. M.—Feng, S. J.: On the Nevanlinna characteristic off(z + η) and difference equations in the complex plane, Ramanujan J. 16 (2008), 105–129.10.1007/s11139-007-9101-1Search in Google Scholar

[4] Chen, Z. X.: On growth, zeros and poles of meromorphic solutions of linear and nonlinear difference equations, Sci. China Math. 54 (2011), 2123–2133.10.1007/s11425-011-4265-ySearch in Google Scholar

[5] Chen, Z. X.: Growth and zeros of meromorphic solution of some linear difference equations, J. Math. Anal. Appl. 373 (2011), 235–241.10.1016/j.jmaa.2010.06.049Search in Google Scholar

[6] Halburd, R. G.—Korhonen, R. J.: Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. 31 (2006), 463–478.Search in Google Scholar

[7] Hayman, W.: Meromorphic Functions, Clarendon Press, Oxford, 1964.Search in Google Scholar

[8] Heittokangas, J.—Korhonen, R.—Laine, I.—Rieppo, J.—Zhang, J. L.: Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity, J. Math. Anal. Appl. 355 (2009), 352–363.10.1016/j.jmaa.2009.01.053Search in Google Scholar

[9] Gross, F.: On the equationfn + gn = 1, Bull. Amer. Math. Soc. 72 (1966), 86–88.10.1090/S0002-9904-1966-11429-5Search in Google Scholar

[10] Gross, F.: On the equationfn + gn = hn, Amer. Math. Monthly 73 (1966), 1093–1096.10.2307/2314644Search in Google Scholar

[11] Korhonen, R.: An extension of Picards theorem for meromorphic functions of small hyper-order, J.Math. Anal. Appl. 357 (2009), 244–253.10.1016/j.jmaa.2009.04.011Search in Google Scholar

[12] Li, S.—GAO, Z. S.: Finite order meromorphic solutions of linear difference equations, Proc. Japan Acad. 87 (2011), 73–76.10.3792/pjaa.87.73Search in Google Scholar

[13] Liu, K.—Qi, X. G.: Meromorphic solutions of q-shift difference equations, Ann. Polon. Math. 101 (2011), 215–225.10.4064/ap101-3-2Search in Google Scholar

[14] Laine, I.—Yang, C. C.: Clunie theorem for difference and q-difference polynomials, J. Lond. Math. Soc. (2) 76 (2007), 556–566.10.1112/jlms/jdm073Search in Google Scholar

[15] Nörlund, N. E.: Differenzenrechnung, Springer, Berlin, 1924 (German).10.1007/978-3-642-50824-0Search in Google Scholar

[16] Qi, X. G.—Liu, Y.—Yang, L. Z.: Nevanlinna theory for thef(qz + c) and its applications, Acta Math. Sci. Ser. A Chin. Ed. 33 (2013), 819–828.Search in Google Scholar

[17] Shimomura, S.: Entire solutions of a polynomial difference equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 253–266.Search in Google Scholar

[18] Whittaker, J. M.: Interpolatory Function Theory, Cambridge University Press/Macmillan, New York, 1935.Search in Google Scholar

[19] Yanagihara, N.: Meromorphic solutions of some difference equations, Funkcial. Ekvac. 23 (1980), 309–326.Search in Google Scholar

[20] Yanagihara, N.: Meromorphic solutions of some difference equations II, Funkcial. Ekvac. 24 (1981), 113–124.Search in Google Scholar

[21] Yanagihara, N.: Meromorphic solutions of some difference equations of nth order, Arch. Ration. Mech. Anal. 91 (1985), 169–192.10.1007/BF00276862Search in Google Scholar

[22] Yang, C. C.—Yi, H. X.: Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, Dordrecht-Boston-London, 2003.10.1007/978-94-017-3626-8Search in Google Scholar

[23] Zhang, J. L.—Korhonen, R.: On the Nevanlinna characteristic off(qz) and its applications, J. Math. Anal. Appl. 369 (2010), 537–544.10.1016/j.jmaa.2010.03.038Search in Google Scholar

Received: 2013-4-30
Accepted: 2013-9-2
Published Online: 2016-8-23
Published in Print: 2016-6-1

© 2016 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Research Article
  2. On the vertex-to-edge duality between the Cayley graph and the coset geometry of von Dyck groups
  3. Research Article
  4. Characterization of hereditarily reversible posets
  5. Research Article
  6. Families of sets which can be represented as sublattices of the lattice of convex subsets of a linearly ordered set
  7. Research Article
  8. Diophantine equation X4+Y4 = 2(U4 + V4)
  9. Research Article
  10. On the congruent number problem over integers of cyclic number fields
  11. Research Article
  12. On the Diophantine equation x2 + C= yn for C = 2a3b17c and C = 2a13b17c
  13. Research Article
  14. Representations and evaluations of the error term in a certain divisor problem
  15. Research Article
  16. A construction of a Peano curve
  17. Research Article
  18. Compositions of ϱ-upper continuous functions
  19. Research Article
  20. Products of Świa̧tkowski functions
  21. Research Article
  22. Subclasses of meromorphically p-valent functions involving a certain linear operator
  23. Research Article
  24. Oscillation results for second-order mixed neutral differential equations with distributed deviating arguments
  25. Research Article
  26. Oscillation criteria for higher order nonlinear delay dynamic equations on time scales
  27. Research Article
  28. On the solutions of a fourth order parabolic equation modeling epitaxial thin film growth
  29. Research Article
  30. Meromorphic solutions of q-shift difference equations
  31. Research Article
  32. Oscillation criteria for higher-order nonlinear delay dynamic equations on time scales
  33. Research Article
  34. Multiplier spaces and the summing operator for series
  35. Research Article
  36. Weighted αβ-statistical convergence of Kantorovich-Mittag-Leffler operators
  37. Research Article
  38. The local spectral radius of a nonnegative orbit of compact linear operators
  39. Research Article
  40. A note on Lie product preserving maps on Mn(ℝ)
  41. Research Article
  42. On vector optimality conditions for constrained problems with ℓ-stable data
  43. Research Article
  44. Reparameterization of weakly nonlinear regression models with constraints
  45. Research Article
  46. Convolutions of polynomial kernels
  47. Research Article
  48. Closed form evaluation of sums containing squares of Fibonomial coefficients
https://doi.org/10.1515/ms-2015-0168
Keywords for this article
meromorphic functions; growth; q-shift difference equation

Articles in the same Issue

  1. Research Article
  2. On the vertex-to-edge duality between the Cayley graph and the coset geometry of von Dyck groups
  3. Research Article
  4. Characterization of hereditarily reversible posets
  5. Research Article
  6. Families of sets which can be represented as sublattices of the lattice of convex subsets of a linearly ordered set
  7. Research Article
  8. Diophantine equation X4+Y4 = 2(U4 + V4)
  9. Research Article
  10. On the congruent number problem over integers of cyclic number fields
  11. Research Article
  12. On the Diophantine equation x2 + C= yn for C = 2a3b17c and C = 2a13b17c
  13. Research Article
  14. Representations and evaluations of the error term in a certain divisor problem
  15. Research Article
  16. A construction of a Peano curve
  17. Research Article
  18. Compositions of ϱ-upper continuous functions
  19. Research Article
  20. Products of Świa̧tkowski functions
  21. Research Article
  22. Subclasses of meromorphically p-valent functions involving a certain linear operator
  23. Research Article
  24. Oscillation results for second-order mixed neutral differential equations with distributed deviating arguments
  25. Research Article
  26. Oscillation criteria for higher order nonlinear delay dynamic equations on time scales
  27. Research Article
  28. On the solutions of a fourth order parabolic equation modeling epitaxial thin film growth
  29. Research Article
  30. Meromorphic solutions of q-shift difference equations
  31. Research Article
  32. Oscillation criteria for higher-order nonlinear delay dynamic equations on time scales
  33. Research Article
  34. Multiplier spaces and the summing operator for series
  35. Research Article
  36. Weighted αβ-statistical convergence of Kantorovich-Mittag-Leffler operators
  37. Research Article
  38. The local spectral radius of a nonnegative orbit of compact linear operators
  39. Research Article
  40. A note on Lie product preserving maps on Mn(ℝ)
  41. Research Article
  42. On vector optimality conditions for constrained problems with ℓ-stable data
  43. Research Article
  44. Reparameterization of weakly nonlinear regression models with constraints
  45. Research Article
  46. Convolutions of polynomial kernels
  47. Research Article
  48. Closed form evaluation of sums containing squares of Fibonomial coefficients
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 16.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2015-0168/pdf
Scroll to top button