On Complex Differential-Difference Equations of Malmquist Type

Abstract

In this paper, we shall give some properties of complex differential-difference equation of the form $\left({\displaystyle {\sum }_{\mu =1}^n{\alpha }_{\mu }}\left(z\right){\displaystyle {\prod }_{i=1}^{t}f}{\left(z+c_i\right)}^{a_{i0}^{\mu }}{\left(f^{\prime }\left(z+c_i\right)\right)}^{a_{i1}^{\mu }}\cdots {\left(f^{\left(k_i\right)}\left(z+c_i\right)\right)}^{a_{i{k}_i}^{\mu }}\right)/\left({\displaystyle {\sum }_{\nu =1}^m{\beta }_{\nu }}\left(z\right){\displaystyle {\prod }_{i=1}^{t}f}{\left(z+c_i\right)}^{b_{i0}^{\nu }}{\left(f^{\prime }\left(z+c_i\right)\right)}^{b_{i1}^{\nu }}\cdots {\left(f^{\left(k_i\right)}\left(z+c_i\right)\right)}^{b_{i{k}_i}^{\nu }}\right)=R\left(z,f\left(z\right)\right),$ where $a_{ij}^{\mu }\left(\mu =\mathrm{1,2},\dots ,n\right)$ and $b_{ij}^{\nu }\left(\nu =\mathrm{1,2},\dots ,m\right)\left(i=\mathrm{1,2},\dots ,t;j=\mathrm{0,1},\dots ,k_i\right)$ are non-negative integers, $c_i\left(i=\mathrm{1,2},\dots ,t\right)$ are distinct, nonzero complex numbers, ${\alpha }_{\mu }\left(z\right)\left(\mu =\mathrm{1,2},\dots ,n\right)$ and ${\beta }_{\nu }\left(z\right)\left(\nu =\mathrm{1,2},\dots ,m\right)$ are small functions relative to $f\left(z\right)$ , and $R\left(z,f\left(z\right)\right)$ is a rational function in $f\left(z\right)$ with coefficients, which are small functions of $f\left(z\right)$ . We also consider related complex functional equations in the paper. These results are improvements of some several previous results.