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We describe a uniform way of obtaining basic hypergeometric functions aslimits of the elliptic beta integral. This description gives rise to theconstruction of a polytope with a different basic hypergeometric functionattached to each face of this polytope. We can subsequently obtain variousrelations, such as transformations and three-term relations, of these functionsby considering geometrical properties of this polytope. The most generalfunctions we describe in this way are sums of two very-well-poised${}_{10}\phi_9$'s and their Nassrallah-Rahman type integral representation.

Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions