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

TL;DR

This paper characterizes meromorphic solutions of a nonlinear difference equation of Tumura-Clunie type with order less than one, using Nevanlinna theory, and extends previous results to differential-difference polynomials.

Contribution

It provides a new characterization of solutions in terms of exponential functions and extends prior work to include differential-difference polynomials under certain conditions.

Findings

Solutions are expressed as exponential functions under specified conditions.

Results improve upon previous findings by Chen et al.

Extensions include differential-difference polynomials with additional growth conditions.

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^{{\lambda_1}z}+p_2e^{{\lambda_2}z}+p_3e^{{\lambda_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 $\lambda_j,~p_j\not=0$ 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.