### Abstract

A τ function formalism for Sakai's elliptic Painlevé equation is presented. This establishes the equivalence between the two formulations by Sakai and by Ohta-Ramani-Grammaticos. We also give a simple geometric description of the elliptic Painlevé equation as a non-autonomous deformation of the addition formula on elliptic curves. By using these formulations, we construct a particular solution of die elliptic Painlevé equation expressed in terms of the elliptic hypergeometric function _{10}E_{9}.

Original language | English |
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Pages (from-to) | L263-L272 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 36 |

Issue number | 17 |

DOIs | |

Publication status | Published - May 2 2003 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)

### Cite this

*Journal of Physics A: Mathematical and General*,

*36*(17), L263-L272. https://doi.org/10.1088/0305-4470/36/17/102

**10E9 solution to the elliptic Painlevé equation.** / Kajiwara, Kenji; Masuda, Tetsu; Noumi, Masatoshi; Ohta, Yasuhiro; Yamada, Yasuhiko.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and General*, vol. 36, no. 17, pp. L263-L272. https://doi.org/10.1088/0305-4470/36/17/102

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TY - JOUR

T1 - 10E9 solution to the elliptic Painlevé equation

AU - Kajiwara, Kenji

AU - Masuda, Tetsu

AU - Noumi, Masatoshi

AU - Ohta, Yasuhiro

AU - Yamada, Yasuhiko

PY - 2003/5/2

Y1 - 2003/5/2

N2 - A τ function formalism for Sakai's elliptic Painlevé equation is presented. This establishes the equivalence between the two formulations by Sakai and by Ohta-Ramani-Grammaticos. We also give a simple geometric description of the elliptic Painlevé equation as a non-autonomous deformation of the addition formula on elliptic curves. By using these formulations, we construct a particular solution of die elliptic Painlevé equation expressed in terms of the elliptic hypergeometric function 10E9.

AB - A τ function formalism for Sakai's elliptic Painlevé equation is presented. This establishes the equivalence between the two formulations by Sakai and by Ohta-Ramani-Grammaticos. We also give a simple geometric description of the elliptic Painlevé equation as a non-autonomous deformation of the addition formula on elliptic curves. By using these formulations, we construct a particular solution of die elliptic Painlevé equation expressed in terms of the elliptic hypergeometric function 10E9.

UR - http://www.scopus.com/inward/record.url?scp=0037724054&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037724054&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/36/17/102

DO - 10.1088/0305-4470/36/17/102

M3 - Article

AN - SCOPUS:0037724054

VL - 36

SP - L263-L272

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 17

ER -