### 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 |

### Fingerprint

### Keywords

- Difference Painlevé equation
- Rational solution

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Tokyo Journal of Mathematics*,

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

**Rational solutions of difference painlevé equations.** / Shimomura, Shun.

Research output: Contribution to journal › Article

*Tokyo Journal of Mathematics*, vol. 35, no. 1, pp. 85-95. https://doi.org/10.3836/tjm/1342701346

}

TY - JOUR

T1 - Rational solutions of difference painlevé equations

AU - Shimomura, Shun

PY - 2012/6

Y1 - 2012/6

N2 - We capture all the rational solutions of some difference Painlevé equations of PI and PII types. For non-autonomous cases, it is shown that all the rational solutions of the difference PII are ones generated by successive application of auto-Bäcklund transformations to the seed solution vanishing identically, and that the other equations of PI 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.

AB - We capture all the rational solutions of some difference Painlevé equations of PI and PII types. For non-autonomous cases, it is shown that all the rational solutions of the difference PII are ones generated by successive application of auto-Bäcklund transformations to the seed solution vanishing identically, and that the other equations of PI 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.

KW - Difference Painlevé equation

KW - Rational solution

UR - http://www.scopus.com/inward/record.url?scp=84866927546&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866927546&partnerID=8YFLogxK

U2 - 10.3836/tjm/1342701346

DO - 10.3836/tjm/1342701346

M3 - Article

AN - SCOPUS:84866927546

VL - 35

SP - 85

EP - 95

JO - Tokyo Journal of Mathematics

JF - Tokyo Journal of Mathematics

SN - 0387-3870

IS - 1

ER -