The Endless Beta Integrals

Gor A. Sarkissian ; Vyacheslav P. Spiridonov

Symmetry, integrability and geometry: methods and applications, Tome 16 (2020)

Résumé

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 $\mathrm{SL}(2,\mathbb{C})$ 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 $b\to 1$).

Détail

Keywords: elliptic hypergeometric functions, complex gamma function, beta integrals, star-triangle relation.

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