_10E_9 solution to the elliptic Painlev'e equation

Details _10E_9 solution to the elliptic Painlev'e equation _10E_9 solution to the elliptic Painlev'e equation

2003-03-18

arxiv.org/abs/nlin/0303032v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

12 pages

Scientific paper

A $\tau$ function formalism for Sakai's elliptic Painlev'e 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 Painlev'e equation as a non-autonomous deformation of the addition formula on elliptic curves. By using these formulations, we construct a particular solution of the elliptic Painlev'e equation expressed in terms of the elliptic hypergeometric function_10E_9.

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