On Meromorphic Solutions to a Difference Equation of Tumura-Clunie Type

Abstract

The meromorphic solutions $f$ with ${\rho }_2(f)<1$ of the non-linear difference equation \begin{align*} f^n(z)+P_d(z,f)=p_1e^{{λ_1}z}+p_2e^{{λ_2}z}+p_3e^{{λ_3}z}, \end{align*} are characterized in terms of exponential functions using Nevanlinna theory, under certain conditions on ${\lambda }_j$ for $j=1,2,3$. Here, $n>2$, $P_d(z,f)$ is a difference polynomial in $f$ of degree $\le n-1$, and ${\lambda }_j,p_j\ne 0$ for $j=1,2,3$. These results improve upon those previously obtained by Chen et al.[Bull. Korean Math. Soc. 61, 745-762 (2024)]. Some examples are provided to illustrate these results. Additionally, if $P_d(z,f)$ is a differential-difference polynomial, then under the supplementary condition $N(r,f)=S(r,f)$, by applying the same proof method, these conclusions still hold.