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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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Table of 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.