# 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} }

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…

## 10 Citations

### On Complex Gamma-Function Integrals

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
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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.…

### Barnes-Ismagilov Integrals and Hypergeometric Functions of the Complex Field

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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…

### Introduction to the Theory of Elliptic Hypergeometric Integrals

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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…

### Rational Hypergeometric Identities

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### Lens Partition Functions and Integrability Properties

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: 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…

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- MathematicsJournal of Physics A: Mathematical and Theoretical
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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…

### ELLIPTIC HYPERGEOMETRIC FUNCTION AND 6 j -SYMBOLS FOR THE SL (2 , C ) GROUP

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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…

### Elliptic hypergeometric function and $$6j$$-symbols for the $$SL(2,{\mathbb C})$$ group

- MathematicsTheoretical and Mathematical Physics
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. 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…

### On the Dotsenko-Fateev complex twin of the Selberg integral and its extensions

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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…

### A parafermionic hypergeometric function and supersymmetric 6j-symbols

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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…

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Abstract
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