The Endless Beta Integrals

Abstract

We consider a special degeneration limit ${\omega }_1\to -{\omega }_2$ (or $b\to \mathrm{i}$ 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 $b\to 1$).

equations (62), (69) and (72) corrected

References in corpus (3)

• Essays on the theory of elliptic hypergeometric functions
• Properties of generalized univariate hypergeometric functions
• Exact solution of the Faddeev-Volkov model

Cited by in corpus (8)

• Complex hypergeometric functions and integrable many-body problems
• A parafermionic hypergeometric function and supersymmetric 6j-symbols
• Elliptic hypergeometric function and $6j$-symbols for the SL(2,$\mathbb{C}$) group
• On the Dotsenko-Fateev complex twin of the Selberg integral and its extensions
• Rational hypergeometric identities
• Complex and rational hypergeometric functions on root systems
• From hyperbolic to complex Euler integrals
• BC-type open $SL(2,\mathbb{C})$ spin chain