SIGMA 5 (2009), 059, 31 pages      arXiv:0902.0621      https://doi.org/10.3842/SIGMA.2009.059
Contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions”

Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions

Fokko J. van de Bult and Eric M. Rains
MC 253-37, California Institute of Technology, 91125, Pasadena, CA, USA

Received February 01, 2009; Published online June 10, 2009; Proposition 4.3 corrected March 02, 2018

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 ₁₀φ₉'s and their Nassrallah-Rahman type integral representation.

Key words: elliptic hypergeometric functions, basic hypergeometric functions, transformation formulas.

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