On Meromorphic Solutions to a Difference Equation of Tumura-Clunie Type
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908431891300352 |
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| author | Long, Jianren Xiang, Xuxu |
| author_facet | Long, Jianren Xiang, Xuxu |
| contents | The meromorphic solutions $f$ with $ρ_2(f)<1$ 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 $λ_j$ for $j=1,2,3$. Here, $n>2$, $P_d(z,f)$ is a difference polynomial in $f$ of degree $\le n-1$, and $λ_j,~p_j\not=0$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_02572 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Meromorphic Solutions to a Difference Equation of Tumura-Clunie Type Long, Jianren Xiang, Xuxu Complex Variables 39A45, 30D35 The meromorphic solutions $f$ with $ρ_2(f)<1$ 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 $λ_j$ for $j=1,2,3$. Here, $n>2$, $P_d(z,f)$ is a difference polynomial in $f$ of degree $\le n-1$, and $λ_j,~p_j\not=0$ 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. |
| title | On Meromorphic Solutions to a Difference Equation of Tumura-Clunie Type |
| topic | Complex Variables 39A45, 30D35 |
| url | https://arxiv.org/abs/2507.02572 |