Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions

Abstract

We describe a uniform way of obtaining basic hypergeometric functions as limits of the elliptic beta integral. This description gives rise to the construction of a polytope with a different basic hypergeometric function attached to each face of this polytope. We can subsequently obtain various relations, such as transformations and three-term relations, of these functions by considering geometrical properties of this polytope. The most general functions we describe in this way are sums of two very-well-poised ${}_{10}ϕ_9$'s and their Nassrallah-Rahman type integral representation.

v3: Proposition 4.3 corrected