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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.02243 |
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| _version_ | 1866913774866268160 |
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| author | Liu, Xiong-Feng Wen, Zhi-Tao Zhu, Can-Xin |
| author_facet | Liu, Xiong-Feng Wen, Zhi-Tao Zhu, Can-Xin |
| contents | In this paper, we study the growth of transcendental entire solutions of linear difference equations
\begin{equation}
P_m(z)Δ^mf(z)+\cdots+P_1(z)Δf(z)+P_0(z)f(z)=0,\tag{+}
\end{equation} where $P_j(z)$ are polynomials for $j=0,\ldots,m$. At first, we reveal type of binomial series in terms of its coefficients. Second, we give a list of all possible orders, which are less than 1, and types of transcendental entire solutions of linear difference equations $(+)$. In particular, we give so far the best precise growth estimate of transcendental entire solutions of order less than 1 of $(+)$, which improves results in [3, 4], [5], [7]. Third, for any given rational number $ρ\in(0,1)$ and real number $σ\in(0,\infty)$, we can construct a linear difference equation with polynomial coefficients which has a transcendental entire solution of order $ρ$ and type $σ$. At last, some examples are illustrated for our main theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02243 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The growth of transcendental entire solutions of linear difference equations with polynomial coefficients Liu, Xiong-Feng Wen, Zhi-Tao Zhu, Can-Xin Complex Variables In this paper, we study the growth of transcendental entire solutions of linear difference equations \begin{equation} P_m(z)Δ^mf(z)+\cdots+P_1(z)Δf(z)+P_0(z)f(z)=0,\tag{+} \end{equation} where $P_j(z)$ are polynomials for $j=0,\ldots,m$. At first, we reveal type of binomial series in terms of its coefficients. Second, we give a list of all possible orders, which are less than 1, and types of transcendental entire solutions of linear difference equations $(+)$. In particular, we give so far the best precise growth estimate of transcendental entire solutions of order less than 1 of $(+)$, which improves results in [3, 4], [5], [7]. Third, for any given rational number $ρ\in(0,1)$ and real number $σ\in(0,\infty)$, we can construct a linear difference equation with polynomial coefficients which has a transcendental entire solution of order $ρ$ and type $σ$. At last, some examples are illustrated for our main theorem. |
| title | The growth of transcendental entire solutions of linear difference equations with polynomial coefficients |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2504.02243 |