The Endless Beta Integrals

Abstract

We consider a special degeneration limit ω₁ → −ω₂ (or 𝘣 → 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(𝐸₇) group of symmetry transformations. Resulting functions are identified as hypergeometric functions over the field of complex numbers related to the SL(2, ℂ) group. A new, similar nontrivial hypergeometric degeneration of the Faddeev modular quantum dilogarithm (or hyperbolic gamma function) is discovered in the limit ω₁ → ω₂ (or 𝘣 → 1).