Meromorphic solutions of a complex difference equation of Malmquist type
Bull. Korean Math. Soc. 2014 Vol. 51, No. 6, 1735-1748
https://doi.org/10.4134/BKMS.2014.51.6.1735
Published online November 30, 2014
Ran-Ran Zhang and Zhi-Bo Huang
Guangdong University of Education, South China Normal University
Abstract : In this paper, we investigate the finite order transcendental meromorphic solutions of complex difference equation of Malmquist type $$ \prod_{i=1}^nf(z+c_i)=R(z, f), $$ where $c_1$, $\ldots$, $c_n$ $\in\mathbb{C}\setminus\{0\}$, and $R(z, f)$ is an irreducible rational function in $f(z)$ with meromorphic coefficients. We obtain some results on deficiencies of the solutions. Using these results, we prove that the growth order of the finite order solution $f(z)$ is 1, if $f(z)$ has Borel exceptional values $a(\in\mathbb{C})$ and $\infty$. Moreover, we give the forms of $f(z)$.
Keywords : difference equation, meromorphic function, deficiency
MSC numbers : 30D35, 39A10
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