Proceedings of the Japan Academy, Series A, Mathematical Sciences

Finite order meromorphic solutions of linear difference equations

Sheng Li and Zong-Sheng Gao

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In this paper, we mainly investigate the growth and the value distribution of meromorphic solutions of the linear difference equation \begin{equation*} a_{n}(z)f(z+n)+…+a_{1}(z)f(z+1)+a_{0}(z)f(z)=b(z), \end{equation*} where $a_{0}(z),a_{1}(z),\cdots,a_{n}(z),b(z)$ are entire functions such that $a_{0}(z)a_{n}(z)\not\equiv 0$. For a finite order meromorphic solution $f(z)$, some interesting results on the relation between $\rho=\rho(f)$ and $\lambda_{f}=\max\{\lambda(f),\lambda(1/f)\}$, are proved. And examples are provided for our results.

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Proc. Japan Acad. Ser. A Math. Sci., Volume 87, Number 5 (2011), 73-76.

First available in Project Euclid: 26 April 2011

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Primary: 30D35: Distribution of values, Nevanlinna theory 39A13: Difference equations, scaling ($q$-differences) [See also 33Dxx] 39A22: Growth, boundedness, comparison of solutions

Difference equations value distribution finite order


Li, Sheng; Gao, Zong-Sheng. Finite order meromorphic solutions of linear difference equations. Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 5, 73--76. doi:10.3792/pjaa.87.73.

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  • M. J. Ablowitz, R. Halburd and B. Herbst, On the extension of the Painlevé property to difference equations, Nonlinearity 13 (2000), no. 3, 889–905.
  • S. B. Bank and R. P. Kaufman, An extension of Hölder's theorem concerning the gamma function, Funkcial. Ekvac. 19 (1976), no. 1, 53–63.
  • W. Bergweiler and J. K. Langley, Zeros of differences of meromorphic functions, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 1, 133–147.
  • Y.-M. Chiang and S.-J. Feng, On the Nevanlinna characteristic of $f(z+\eta)$ and difference equations in the complex plane, Ramanujan J. 16 (2008), no. 1, 105–129.
  • Y.-M. Chiang and S.-J. Feng, On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions, Trans. Amer. Math. Soc. 361 (2009), no. 7, 3767–3791.
  • R. G. Halburd and R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl. 314 (2006), no. 2, 477–487.
  • R. G. Halburd and R. J. Korhonen, Existence of finite-order meromorphic solutions as a detector of integrability in difference equations, Phys. D 218 (2006), no. 2, 191–203.
  • W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs Clarendon Press, Oxford, 1964.
  • J. Heittokangas et al., Complex difference equations of Malmquist type, Comput. Methods Funct. Theory 1 (2001), no. 1, 27–39.
  • I. Laine, Nevanlinna theory and complex differential equations, de Gruyter Studies in Mathematics, 15, de Gruyter, Berlin, 1993.
  • I. Laine and C.-C. Yang, Clunie theorems for difference and $q$-difference polynomials, J. Lond. Math. Soc. (2) 76 (2007), no. 3, 556–566.
  • C.-C. Yang and I. Laine, On analogies between nonlinear difference and differential equations, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 1, 10–14.
  • M. Ozawa, On the existence of prime periodic entire functions, Kōdai Math. Sem. Rep. 29 (1977/78), no. 3, 308–321.
  • S. Shimomura, Entire solutions of a polynomial difference equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 2, 253–266.
  • J. M. Whittaker, Interpolatory Function Theory. Cambridge University Press, Cambridge, 1935.
  • N. Yanagihara, Meromorphic solutions of some difference equations, Funkcial. Ekvac. 23 (1980), no. 3, 309–326.
  • C.-C. Yang and H.-X. Yi, Uniqueness theory of meromorphic functions, Mathematics and its Applications, 557, Kluwer Acad. Publ., Dordrecht, 2003.