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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2511.01660 |
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| _version_ | 1866918183789658112 |
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| author | Liu, Wenlong |
| author_facet | Liu, Wenlong |
| contents | In this paper, we use the Banach fixed point theorem to examine the existence of meromorphic solutions to the following first-order $q$-difference equation \begin{align}\tag{†}\label{dagger} y(qz)=\frac{a_1(z)y(z)+a_2(z)y(z)^2+\dots+a_p(z)y(z)^p}{1+b_1(z)y(z)+\cdots +b_t(z)y(z)^t}, \end{align} where $q\in \mathbb{C},$ $a_1(z), \dots, a_p(z); b_1(z), \dots, b_t(z)$ are all meromorphic functions. We establish sufficient conditions ensuring the existence and uniqueness of meromorphic solutions that can be extended to the entire complex plane $\mathbb{C}.$
More precisely, we have the following result. If $\left | q \right |\geq 3 $ and \[|a_1(z)| = \max_{1 \le j \le p} |a_j(z)| \le \frac{1}{|z|}, \quad \max_{1 \le k \le t} |b_k(z)| \le \frac{1}{|z|}, \quad z \in \{\, |\Re(z)| \ge ρ> 0 \,\}, \] and $y(0)\ne \infty,$ then we prove that~\eqref{dagger} admits a unique meromorphic solution in $D(ρ),$ which can be extended meromorphically to $\mathbb {C}.$ Moreover, if $a_1(z)\equiv 0,$ the conclusion still holds. Furthermore, if $\left | q \right |\geq 6$ and \begin{gather*} |a_1(z)| \le \frac{1}{|q|}, \quad |a_j(z)| \le |q|^{|z|} \quad (2 \le j \le p), \quad |b_k(z)| \le |q|^{|z|} \quad (1 \le k \le t), \\[4pt] z \in D(ρ,σ)
= \{\, z : |\Re(z)| \le ρ,\; |\Im(z)| \le σ, \,\, ρ>0,\,\, σ>0 \,\}, \end{gather*} and $y(0)\ne \infty,$ then we prove that \eqref{dagger} admits a unique meromorphic solution in $D(ρ, σ),$ which can also be extended meromorphically to $\mathbb {C}.$ This conclusion remains valid in the case where $a_1(z)\equiv 0.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01660 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the extension of analytic solutions of a class of first-order q-difference equations Liu, Wenlong Complex Variables In this paper, we use the Banach fixed point theorem to examine the existence of meromorphic solutions to the following first-order $q$-difference equation \begin{align}\tag{†}\label{dagger} y(qz)=\frac{a_1(z)y(z)+a_2(z)y(z)^2+\dots+a_p(z)y(z)^p}{1+b_1(z)y(z)+\cdots +b_t(z)y(z)^t}, \end{align} where $q\in \mathbb{C},$ $a_1(z), \dots, a_p(z); b_1(z), \dots, b_t(z)$ are all meromorphic functions. We establish sufficient conditions ensuring the existence and uniqueness of meromorphic solutions that can be extended to the entire complex plane $\mathbb{C}.$ More precisely, we have the following result. If $\left | q \right |\geq 3 $ and \[|a_1(z)| = \max_{1 \le j \le p} |a_j(z)| \le \frac{1}{|z|}, \quad \max_{1 \le k \le t} |b_k(z)| \le \frac{1}{|z|}, \quad z \in \{\, |\Re(z)| \ge ρ> 0 \,\}, \] and $y(0)\ne \infty,$ then we prove that~\eqref{dagger} admits a unique meromorphic solution in $D(ρ),$ which can be extended meromorphically to $\mathbb {C}.$ Moreover, if $a_1(z)\equiv 0,$ the conclusion still holds. Furthermore, if $\left | q \right |\geq 6$ and \begin{gather*} |a_1(z)| \le \frac{1}{|q|}, \quad |a_j(z)| \le |q|^{|z|} \quad (2 \le j \le p), \quad |b_k(z)| \le |q|^{|z|} \quad (1 \le k \le t), \\[4pt] z \in D(ρ,σ) = \{\, z : |\Re(z)| \le ρ,\; |\Im(z)| \le σ, \,\, ρ>0,\,\, σ>0 \,\}, \end{gather*} and $y(0)\ne \infty,$ then we prove that \eqref{dagger} admits a unique meromorphic solution in $D(ρ, σ),$ which can also be extended meromorphically to $\mathbb {C}.$ This conclusion remains valid in the case where $a_1(z)\equiv 0.$ |
| title | On the extension of analytic solutions of a class of first-order q-difference equations |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2511.01660 |