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

SIGMA 16 (2020), 074, 21 pages      arXiv:2005.01059      https://doi.org/10.3842/SIGMA.2020.074
Contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quantum Field Theory

The Endless Beta Integrals

Gor A. Sarkissian abc and Vyacheslav P. Spiridonov ac
a)  Laboratory of Theoretical Physics, JINR, Dubna, 141980, Russia
b)  Department of Physics, Yerevan State University, Yerevan, Armenia
c)  St. Petersburg Department of the Steklov Mathematical Institute of Russian Academy of Sciences, Fontanka 27, St. Petersburg, 191023 Russia

Received May 05, 2020, in final form July 24, 2020; Published online August 05, 2020; Equations (62), (69) and (72) corrected October 22, 2021

Abstract
We consider a special degeneration limit $\omega_1\to - \omega_2$ (or $b\to {\rm i}$ in the context of $2d$ Liouville quantum field theory) for the most general univariate hyperbolic beta integral. This limit is also applied to the most general hyperbolic analogue of the Euler-Gauss hypergeometric function and its $W(E_7)$ group of symmetry transformations. Resulting functions are identified as hypergeometric functions over the field of complex numbers related to the ${\rm SL}(2,\mathbb{C})$ group. A new similar nontrivial hypergeometric degeneration of the Faddeev modular quantum dilogarithm (or hyperbolic gamma function) is discovered in the limit $\omega_1\to \omega_2$ (or $b\to 1$).

Key words: elliptic hypergeometric functions; complex gamma function; beta integrals; star-triangle relation.

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