# 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}
}
• 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 ${\rm SL}(2,\mathbb{C… • Mathematics Symmetry, Integrability and Geometry: Methods and Applications • 2020 It was observed recently that relations between matrix elements of certain operators in the${\rm SL}(2,\mathbb R)$spin chain models take the form of multidimensional integrals derived by R.A. We examine a family${}_pG_{q}^{\mathbb C}\big[\genfrac{}{}{0pt}{}{(a)}{(b)};z\big]$of integrals of Mellin-Barnes type over the space${\mathbb Z}\times {\mathbb R}$, such functions$G$naturally We give a brief account of the key properties of elliptic hypergeometric integrals—a relatively recently discovered top class of transcendental special functions of hypergeometric type. In • Mathematics Functional Analysis and Its Applications • 2021 A special singular limit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} • Physics, Mathematics • 2021 : We study lens partitions functions for the three-dimensional N = 2 supersymmetric gauge theories on S 3 b / Z r . We consider an equality as a new hyperbolic hypergeometric solution to the • Mathematics Journal of Physics A: Mathematical and Theoretical • 2022 General reduction of the elliptic hypergeometric equation to the level of complex hypergeometric functions is described. The derived equation is generalized to the Hamiltonian eigenvalue problem for • Mathematics • 2022 We show that the complex hypergeometric function describing 6 j -symbols for the SL (2 , C ) group is a special degeneration of the V -function—an elliptic analogue of the Euler–Gauss 2 F 1 • Mathematics Theoretical and Mathematical Physics • 2022 . We show that the complex hypergeometric function describing 6 j -symbols for SL (2 , C ) group is a special degeneration of the V -function — an elliptic analogue of the Euler-Gauss 2 F 1 The complex Dotsenko-Fateev integral I n ( σ, τ ; θ ) is the Selberg beta integral, where real variables in the integrand Q | x k | σ − 1 | 1 − x k | τ − 1 Q | x k − x l | 2 θ are replaced by complex • Mathematics • 2022 We study properties of a parafermionic generalization of the hyperbolic hypergeometric function, appearing as the most important part in the fusion matrix for Liouville ﬁeld theory and the ## References SHOWING 1-10 OF 69 REFERENCES • Mathematics Symmetry, Integrability and Geometry: Methods and Applications • 2020 It was observed recently that relations between matrix elements of certain operators in the${\rm SL}(2,\mathbb R)$spin chain models take the form of multidimensional integrals derived by R.A. • Mathematics • 2016 We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum\slash integrals associated to the$A_n$and$BC_n$root systems, generalising the formulas previously obtained We examine a family${}_pG_{q}^{\mathbb C}\big[\genfrac{}{}{0pt}{}{(a)}{(b)};z\big]$of integrals of Mellin-Barnes type over the space${\mathbb Z}\times {\mathbb R}$, such functions$G\$ naturally
We define a general class of (multiple) integrals of hypergeometric type associated with the Jacobi theta functions. These integrals are related to theta hypergeometric series through the residue
We derive a beta-integral over $${\mathbb {Z}}\times {\mathbb {R}}$$ Z × R , which is a counterpart of the Dougall $$_5H_5$$ 5 H 5 -formula and of the de Branges–Wilson integral, our integral
Abstract In Ann. Math., to appear, 2008, the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various
The univariate elliptic beta integral was discovered by the author in 2000. Recently Bazhanov and Sergeev have interpreted it as a star-triangle relation (STR). This important observation is
• Mathematics
• 2007
Based on Spiridonov’s analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and