Duality relations for hypergeometric series

We explicitly give the relations between the hypergeometric solutions of the general hypergeometric equation and their duals, as well as similar relations for $q$-hypergeometric equations. They form a family of very general identities for hypergeometric series. Although they were foreseen already by Bailey in the 1930s on analytic grounds, we give a purely algebraic treatment based on general principles in general differential and difference modules.