### Abstract

The discrete second Painlevé equation dP _{II} is mapped to the second Painlevé equation P _{II} by its continuous limit, and then, as shown by Kajiwara et al., a rational solution of dP _{II} also reduces to that of P _{II}. In this paper, regarding dP _{II} as a difference equation, we present a certain asymptotic solution that reduces to a triply-truncated solution of P _{II} in this continuous limit. In a special case our solution corresponds to a rational one of dP _{II}. Furthermore we show the existence of families of solutions having sequential limits to truncated solutions of P _{II}.

Original language | English |
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Pages (from-to) | 733-781 |

Number of pages | 49 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 64 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2012 Oct 5 |

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### Keywords

- Asymptotic solution
- Continuous limit
- Difference second Painlevé equation
- Second Painlevé equation

### ASJC Scopus subject areas

- Mathematics(all)