### Abstract

We capture all the rational solutions of some difference Painlevé equations of P_{I} and P_{II} types. For non-autonomous cases, it is shown that all the rational solutions of the difference P_{II} are ones generated by successive application of auto-Bäcklund transformations to the seed solution vanishing identically, and that the other equations of P_{I} type admit no rational solutions. For autonomous cases, all the nontrivial rational solutions are obtained, and they exist under a certain condition on a fixed point of the equation. If such a condition is not satisfied, there exist solutions that are rational in an exponential function.

Original language | English |
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Pages (from-to) | 85-95 |

Number of pages | 11 |

Journal | Tokyo Journal of Mathematics |

Volume | 35 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2012 Jun 1 |

### Keywords

- Difference Painlevé equation
- Rational solution

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Shimomura, S. (2012). Rational solutions of difference painlevé equations.

*Tokyo Journal of Mathematics*,*35*(1), 85-95. https://doi.org/10.3836/tjm/1342701346