M.P. Appell's function and vector bundles of rank 2 on elliptic curves

This paper is devoted to the function introduced by M.P. Appell in connecti on with decomposition of elliptic functions of the third kind into simple e lements. We show that this function is related to global sections of rank-2 vector bundles on elliptic curves. We derive analogues of theta-identities for this function and prove the divisibility property for the action of th e modular group, that should be considered as a replacement of the function al equation.