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 $\sum \lambda \in I{\alpha }_{\lambda }(z)({\prod }_{j=0}^{n}f{(z+c_j)}^{{\lambda }_j})=R(z,f\circ p)=((a_0(z)+a_1(z)(f\circ p)+\hspace{0.17em}\cdots \hspace{0.17em}+a_s(z)$ $(f\circ p{)}^s)/(b_0(z)+b_1(z)(f\circ p)+\hspace{0.17em}\cdots \hspace{0.17em}+b_t(z)(f\circ p{)}^t))$, where $I$ is a finite set of multi-indexes $\lambda =({\lambda }_{\mathrm{0}},{\lambda }_{\mathrm{1}},\dots ,{\lambda }_n)$, $c_0=0,c_j\in ℂ\setminus \{0\}\hspace{0.17em}(j=\mathrm{1,2},\dots ,n)$ are distinct complex constants, $p(z)$ is a polynomial, and ${\alpha }_{\lambda }(z)(\lambda \in I)$, $a_i(z)(i=\mathrm{0,1},\dots ,s)$, and $b_j(z)(j=\mathrm{0,1},\dots ,t)$ 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.