2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/80433A $τ$ 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_9.12 pagesExactly Solvable and Integrable SystemsQuantum Algebratext