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 ${}_{10}{\varphi }_9$'s and their Nassrallah-Rahman type integral representation.

v3: Proposition 4.3 corrected

References in corpus (3)

• Essays on the theory of elliptic hypergeometric functions
• Properties of generalized univariate hypergeometric functions
• Elliptic Littlewood identities

Cited by in corpus (8)

• An elliptic hypergeometric integral with W(F_4) symmetry
• Limits of elliptic hypergeometric biorthogonal functions
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• Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions
• Elliptic Racah polynomials
• Utility of integral representations for basic hypergeometric functions and orthogonal polynomials
• On a Family of Integrals that extend the Askey-Wilson Integral
• More basic hypergeometric limits of the elliptic hypergeometric beta integral