## Abstract and Applied Analysis

### Some Properties on Complex Functional Difference Equations

#### Abstract

We obtain some results on the transcendental meromorphic solutions of complex functional difference equations of the form $\sum \lambda \in I{\alpha }_{\lambda }(z)({\prod }_{j=0}^{n}f{(z+{c}_{j})}^{{\lambda }_{j}})=R(z,f\circ p)=(({a}_{0}(z)+{a}_{1}(z)(f\circ p)+ \cdots +{a}_{s}(z)$$(f\circ p{)}^{s})/({b}_{0}(z)+{b}_{1}(z)(f\circ p)+ \cdots +{b}_{t}(z)(f\circ p{)}^{t}))$, where $I$ is a finite set of multi-indexes $\lambda =({\lambda }_{\mathrm{0}},{\lambda }_{\mathrm{1}},\dots ,{\lambda }_{n})$, ${c}_{0}=0,{c}_{j}\in \Bbb C\setminus \{0\} (j=1,2,\dots ,n)$ are distinct complex constants, $p(z)$ is a polynomial, and ${\alpha }_{\lambda }(z) (\lambda \in I)$, ${a}_{i}(z) (i=0,1,\dots ,s)$, and ${b}_{j}(z) (j=0,1,\dots ,t)$ are small meromorphic functions relative to $f(z)$. We further investigate the above functional difference equation which has special type if its solution has Borel exceptional zero and pole.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 283895, 10 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412607223

Digital Object Identifier
doi:10.1155/2014/283895

Mathematical Reviews number (MathSciNet)
MR3200774

Zentralblatt MATH identifier
07022088

#### Citation

Huang, Zhi-Bo; Zhang, Ran-Ran. Some Properties on Complex Functional Difference Equations. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 283895, 10 pages. doi:10.1155/2014/283895. https://projecteuclid.org/euclid.aaa/1412607223

#### References

• W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, UK, 1964.
• M. J. Ablowitz, R. Halburd, and B. Herbst, “On the extension of the Painlevé property to difference equations,” Nonlinearity, vol. 13, no. 3, pp. 889–905, 2000.
• B. Grammaticos, T. Tamizhmani, A. Ramani, and K. M. Tamizhmani, “Growth and integrability in discrete systems,” Journal of Physics A, vol. 34, no. 18, pp. 3811–3821, 2001.
• J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and K. Tohge, “Complex difference equations of Malmquist type,” Computational Methods and Function Theory, vol. 1, no. 1, pp. 27–39, 2001.
• I. Laine, J. Rieppo, and H. Silvennoinen, “Remarks on complex difference equations,” Computational Methods and Function Theory, vol. 5, no. 1, pp. 77–88, 2005.
• G. G. Gundersen, J. Heittokangas, I. Laine, J. Rieppo, and D. Yang, “Meromorphic solutions of generalized Schröder equations,” Aequationes Mathematicae, vol. 63, no. 1-2, pp. 110–135, 2002.
• W. G. Kelley and A. C. Peterson, Difference Equations, Academic Press, Boston, Mass, USA, 1991.
• I. Laine and C. C. Yang, “Clunie theorems for difference and $q$-difference polynomials,” Journal of the London Mathematical Society, vol. 76, no. 3, pp. 556–566, 2007.
• R. Goldstein, “Some results on factorisation of meromorphic functions,” Journal of the London Mathematical Society, vol. 4, pp. 357–364, 1971.
• V. I. Gromak, I. Laine, and S. Shimomura, Painlevé Differential Equations in the Complex Plane, vol. 28, Walter de Gruyter, Berlin, Germany, 2002.
• G. G. Gundersen, “Finite order solutions of second order linear differential equations,” Transactions of the American Mathematical Society, vol. 305, no. 1, pp. 415–429, 1988.
• R. Goldstein, “On meromorphic solutions of certain functional equations,” Aequationes Mathematicae, vol. 18, no. 1-2, pp. 112–157, 1978.
• C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, vol. 557, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 2003.
• Z. X. Chen and K. H. Shon, “On growth of meromorphic solutions for linear difference equations,” Abstract and Applied Analysis, vol. 2013, Article ID 619296, 6 pages, 2013.
• G. Jank and L. Volkmann, Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, Birkhäuser, Basel, Switzerland, 1985.
• J. Wang, “Growth and poles of meromorphic solutions of some difference equations,” Journal of Mathematical Analysis and Applications, vol. 379, no. 1, pp. 367–377, 2011.
• R. G. Halburd and R. J. Korhonen, “Finite-order meromorphic solutions and the discrete Painlevé equations,” Proceedings of the London Mathematical Society, vol. 94, no. 2, pp. 443–474, 2007.
• Y. M. Chiang and S. J. Feng, “On the Nevanlinna characteristic of $f(z+\eta )$ and difference equations in the complex plane,” Ramanujan Journal, vol. 16, no. 1, pp. 105–129, 2008.
• R. G. Halburd and R. J. Korhonen, “Difference analogue of the lemma on the logarithmic derivative with applications to difference equations,” Journal of Mathematical Analysis and Applications, vol. 314, no. 2, pp. 477–487, 2006. \endinput