### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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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