Functional Equations for Appell's F₁ Arising from Transformations of Elliptic Curves

Abstract

We give a functional equation for Appell's hypergeometric function F₁, which arises from transformations of elliptic curves. As an application, we give an efficient algorithm for computing incomplete elliptic integrals of the first kind. We also give a reduction formula that simplifies Lauricella's hypergeometric function F_D of five variables to F₁.