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Published October 2007 | Submitted
Journal Article Open

Properties of Generalized Univariate Hypergeometric Functions

Abstract

Based on Spiridonov's analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E_7 (elliptic, hyperbolic) and of type E_6 (trigonometric) using the appropriate versions of the Nassrallah-Rahman beta integral, and we derive contiguous relations using fundamental addition formulas for theta and sine functions. The top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with Ruijsenaars' relativistic hypergeometric function and the Askey-Wilson function, respectively. We show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions. We also describe an intimate connection between the hyperbolic and trigonometric theory, which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions.

Additional Information

© 2007 Springer-Verlag. Received: 4 August 2006; Accepted: 4 January 2007; Published online: 24 July 2007. Rains was supported in part by NSF Grant No. DMS-0401387. Stokman was supported by the Netherlands Organization for Scientific Research (NWO) in the VIDI-project "Symmetry and modularity in exactly solvable models".

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August 19, 2023
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