Lauricella's hypergeometric function $F_D$ is related to (weighted) configurations of points on $\mathbb{P}^1$. I am looking for generalizations to weighted point configurations on an elliptic curve. I can neither find references nor do I get very far in developing such a notion. I can write down an integral representation of a plausible candidate (in terms of products of powers of theta functions, a la Schwarz-Christoffel) but fail to determine any further properties (such as a set of differential equations for example). Any references welcome!