Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions

TL;DR

This paper presents a geometric approach to deriving basic hypergeometric functions as limits of elliptic hypergeometric functions, leading to new relations and a polytope structure connecting different functions.

Contribution

It introduces a novel geometric framework linking elliptic and basic hypergeometric functions through a polytope construction, enabling derivation of transformation and relation formulas.

Findings

Constructed a polytope with hypergeometric functions on its faces

Derived new transformation and relation formulas for these functions

Connected sums of two very-well-poised ${}_{10}\phi_9$'s to geometric properties

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 ${}_{10}\phi_9$'s and their Nassrallah-Rahman type integral representation.