On the extension of analytic solutions of first-order difference equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909480382365696 |
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| author | Halburd, Rod Korhonen, Risto Liu, Yan Meng, Techheang |
| author_facet | Halburd, Rod Korhonen, Risto Liu, Yan Meng, Techheang |
| contents | We will consider first-order difference equations of the form \[ y(z+1) = \frac{λy(z)+a_2(z)y(z)^2+\cdots+a_p(z)y(z)^p}{1 + b_1(z)y(z)+\cdots+b_q(z)y(z)^q}, \]
where $λ\in\mathbb{C}\setminus\{0\}$ and the coefficients $a_j(z)$ and $b_k(z)$ are meromorphic.
When existence of an analytic solution can be proved for large negative values of $\Re(z)$, the equation determines a unique extension to a global meromorphic solution. In this paper we prove the existence of non-constant meromorphic solutions when the coefficients satisfy $|a_{j}(z)|\leq ν^{|z|}$ and $|b_{k}(z)|\leq ν^{|z|}$ for some $ν<|λ|$ in a half-plane. Furthermore, when a solution exists that is analytic for large positive values of $\Re(z)$, the equation determines a unique extension to a global solution that will generically have algebraic branch points. We analyse a particular constant coefficient equation, $y(z+1)=λy(z)+y(z)^2$, $0<λ<1$, and describe in detail the infinitely-sheeted Riemann surface for such a solution. We also describe solutions with natural boundaries found by Mahler. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03955 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the extension of analytic solutions of first-order difference equations Halburd, Rod Korhonen, Risto Liu, Yan Meng, Techheang Complex Variables 30D05 We will consider first-order difference equations of the form \[ y(z+1) = \frac{λy(z)+a_2(z)y(z)^2+\cdots+a_p(z)y(z)^p}{1 + b_1(z)y(z)+\cdots+b_q(z)y(z)^q}, \] where $λ\in\mathbb{C}\setminus\{0\}$ and the coefficients $a_j(z)$ and $b_k(z)$ are meromorphic. When existence of an analytic solution can be proved for large negative values of $\Re(z)$, the equation determines a unique extension to a global meromorphic solution. In this paper we prove the existence of non-constant meromorphic solutions when the coefficients satisfy $|a_{j}(z)|\leq ν^{|z|}$ and $|b_{k}(z)|\leq ν^{|z|}$ for some $ν<|λ|$ in a half-plane. Furthermore, when a solution exists that is analytic for large positive values of $\Re(z)$, the equation determines a unique extension to a global solution that will generically have algebraic branch points. We analyse a particular constant coefficient equation, $y(z+1)=λy(z)+y(z)^2$, $0<λ<1$, and describe in detail the infinitely-sheeted Riemann surface for such a solution. We also describe solutions with natural boundaries found by Mahler. |
| title | On the extension of analytic solutions of first-order difference equations |
| topic | Complex Variables 30D05 |
| url | https://arxiv.org/abs/2502.03955 |