标题： 关于Tumura-Clunie型差分方程的亚纯解 显示英文标题

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

摘要： 亚纯解$f$在$\rho_2(f)<1$条件下，通过使用 Nevanlinna 理论，对非线性差分方程\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*}的解进行了表征，这些条件涉及$\lambda_j$对$j=1,2,3$的影响。 这里，$n>2$，$P_d(z,f)$是关于$f$的次数为$\le n-1$的差分多项式，而$\lambda_j,~p_j\not=0$对应于$~j=1,2,3$。这些结果改进了Chen等人之前获得的结果[Bull. Korean Math. Soc. 61, 745-762 (2024)]。提供了一些例子来说明这些结果。 此外，如果$P_d(z,f)$是一个微分差多项式，则在补充条件$N(r,f)=S(r,f)$下，通过应用相同的证明方法，这些结论仍然成立。 显示英文摘要

摘要： 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.