The endless beta integrals (Q2217949)

In this paper, the authors performed a comprehensive analysis of the degeneration of hyperbolic integrals to complex hypergeometric functions in the Mellin Barnes representation. In addition, in the limit \(w_{1} \to w_{2} \), the authors discovered a new nontrivial degeneration of the hyperbolic gamma function, or Faddeev's modular quantum dilogarithm, which corresponds to \(b \to \pm 1\) in the Liouville theory, resulting in \(c = 25\) as the central charge. The most general hyperbolic analogue of the Euler Gauss hypergeometric function and its \(W(E_{7})\) group of symmetry transformations are also subject to this limit. The resulting functions are identified as hypergeometric functions over the complex number field related to the SL\((2;\mathbb{C})\) group. In the limit, a new nontrivial hypergeometric degeneration of the Faddeev modular quantum dilogarithm (or hyperbolic gamma function) is found \(w_{1} \to w_{2} \) (or \(b \to 1\)).