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.

46 pages

References in corpus (6)

• Recurrences for elliptic hypergeometric integrals
• Ruijsenaars' hypergeometric function and the modular double of U_q(sl(2,C))
• Elliptic U(2) quantum group and elliptic hypergeometric series
• The Wilson function transform
• The Askey-Wilson function transform
• Short proofs of the elliptic beta integrals

Cited by in corpus (25)

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• The Endless Beta Integrals
• A Relativistic Conical Function and its Whittaker Limits
• Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case
• Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions
• Complex hypergeometric functions and integrable many-body problems
• A parafermionic hypergeometric function and supersymmetric 6j-symbols
• Introduction to the theory of elliptic hypergeometric integrals
• Elliptic hypergeometric function and $6j$-symbols for the SL(2,$\mathbb{C}$) group
• Hyperbolic and trigonometric hypergeometric solutions to the star-star equation
• General modular quantum dilogarithm and beta integrals
• Symmetry Groups of $A_n$ Hypergeometric Series
• The Virasoro fusion kernel and Ruijsenaars' hypergeometric function
• Some Remarks on Very-Well-Poised 8phi7 Series
• Coxeter group actions on Saalschützian ${}_4{F}_3(1)$ series and very-well-poised ${}_7{F}_6(1)$ series
• Aspects of elliptic hypergeometric functions
• Generalized Cherednik-Macdonald identities
• Complex rational Ruijsenaars model. The two-particle case
• Relations for a class of terminating ${}_4{F}_3(4)$ hypergeometric series
• On a Family of Integrals that extend the Askey-Wilson Integral
• From hyperbolic to complex Euler integrals
• Relations among complementary and supplementary pairings of Saalschutzian 4F3(1) series