Essays on the theory of elliptic hypergeometric functions

@article{Spiridonov2008EssaysOT,
  title={Essays on the theory of elliptic hypergeometric functions},
  author={Vyacheslav P. Spiridonov},
  journal={Russian Mathematical Surveys},
  year={2008},
  volume={63},
  pages={405-472}
}
  • V. Spiridonov
  • Published 20 May 2008
  • Mathematics
  • Russian Mathematical Surveys
This is a brief survey of the main results of the theory of elliptic hypergeometric functions -- a new class of special functions of mathematical physics. A proof is given of the most general known univariate exact integration formula generalizing Euler's beta integral. It is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its… 
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155 Citations
Highly Influential Citations
3
Background Citations
66
Methods Citations
16
Results Citations
1

155 Citations

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References

SHOWING 1-10 OF 126 REFERENCES

Limits of elliptic hypergeometric integrals

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…

Continuous biorthogonality of the elliptic hypergeometric function

We construct a family of continuous biorthogonal functions related to an elliptic analogue of the Gauss hypergeometric function. The key tools used for that are the elliptic beta integral and the…

Theta hypergeometric integrals

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…

The Elliptic Gamma Function and SL(3, Z)⋉Z3

Abstract The elliptic gamma function is a generalization of the Euler gamma function and is associated to an elliptic curve. Its trigonometric and rational degenerations are the Jackson q-gamma…

Transformations of elliptic hypergeometric integrals

We prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions, corresponding to the root systems BC_n and A_n; as a special case, we recover some integral…

Recurrences for elliptic hypergeometric integrals

In recent work (math.QA/0309252) on multivariate hypergeometric integrals, the author generalized a conjectural integral formula of van Diejen and Spiridonov to a ten parameter integral provably…

Modular Hypergeometric Residue Sums of Elliptic Selberg Integrals

It is shown that the residue expansion of an elliptic Selberg integral gives rise to an integral representation for a multiple modular hypergeometric series. A conjectural evaluation formula for the…

Elements of the theory of elliptic functions

General theorems about elliptic functions Modular functions The Weierstrass functions Theta functions The Jacobi functions Transformation of elliptic functions Additional facts about elliptic…

Elliptic hypergeometric functions and Calogero-Sutherland-type models

We consider an elliptic analogue of the Gauss hypergeometric function and two of its multivariate generalizations. We describe their relation to elliptic beta integrals, the exceptional Weyl group…

Elliptic polynomials orthogonal on the unit circle with a dense point spectrum

We introduce two explicit examples of polynomials orthogonal on the unit circle. Moments and the reflection coefficients are expressed in terms of the Jacobi elliptic functions. We find explicit…
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