• DOI:10.3842/SIGMA.2020.074
• Corpus ID: 218487243

The Endless Beta Integrals

@article{Sarkissian2020TheEB,
  title={The Endless Beta Integrals},
  author={G A Sarkissian and Vyacheslav P. Spiridonov},
  journal={Symmetry, Integrability and Geometry: Methods and Applications},
  year={2020}
}

• G. Sarkissian, V. Spiridonov
• Published 3 May 2020
• Mathematics
• Symmetry, Integrability and Geometry: Methods and Applications

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 $b\to 1$).