Zero order meromorphic solutions of $q$-difference equations of Malmquist type
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909192626896896 |
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| author | Korhonen, Risto Zhang, Yueyang |
| author_facet | Korhonen, Risto Zhang, Yueyang |
| contents | We consider the first order $q$-difference equation \begin{equation}\tag† f(qz)^n=R(z,f), \end{equation} where $q\not=0,1$ is a constant and $R(z,f)$ is rational in both arguments. When $|q|\not=1$, we show that, if $(†)$ has a zero order transcendental meromorphic solution, then $(†)$ reduces to a $q$-difference linear or Riccati equation, or to an equation that can be transformed to a $q$-difference Riccati equation. In the autonomous case, explicit meromorphic solutions of $(†)$ are presented. Given that $(†)$ can be transformed into a difference equation, we proceed to discuss the growth of the composite function $f(ω(z))$, where $ω(z)$ is an entire function satisfying $ω(z+1)=qω(z)$, and demonstrate how the proposed difference Painlevé property, as discussed in the literature, applies for $q$-difference equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03936 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Zero order meromorphic solutions of $q$-difference equations of Malmquist type Korhonen, Risto Zhang, Yueyang Complex Variables Primary 39A13, Secondary 30D35 and 39A12 We consider the first order $q$-difference equation \begin{equation}\tag† f(qz)^n=R(z,f), \end{equation} where $q\not=0,1$ is a constant and $R(z,f)$ is rational in both arguments. When $|q|\not=1$, we show that, if $(†)$ has a zero order transcendental meromorphic solution, then $(†)$ reduces to a $q$-difference linear or Riccati equation, or to an equation that can be transformed to a $q$-difference Riccati equation. In the autonomous case, explicit meromorphic solutions of $(†)$ are presented. Given that $(†)$ can be transformed into a difference equation, we proceed to discuss the growth of the composite function $f(ω(z))$, where $ω(z)$ is an entire function satisfying $ω(z+1)=qω(z)$, and demonstrate how the proposed difference Painlevé property, as discussed in the literature, applies for $q$-difference equations. |
| title | Zero order meromorphic solutions of $q$-difference equations of Malmquist type |
| topic | Complex Variables Primary 39A13, Secondary 30D35 and 39A12 |
| url | https://arxiv.org/abs/2405.03936 |