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This is a brief overview of the status of the theory of elliptic hypergeometric functions to the end of 2012 written as a complementary chapter to the Russian edition of the book by G.E.
E. W. Barnes, On the theory of the multiple gamma function
1904
Earlier work this paper cites.
F. H. Jackson, The basic gamma-function and the elliptic functions
1905
Earlier work this paper cites.
N. I. Akhiezer, Elements of the theory of elliptic functions
1970
Earlier work this paper cites.
R. J. Baxter, Partition function of the eight-vertex lattice model
1972
Earlier work this paper cites.
by same author, An elliptic incarnation of the Bailey chain
1977
Earlier work this paper cites.
D. M. Bressoud, A matrix inverse
1983
Earlier work this paper cites.
E. K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation. Representation of a quantum algebra
1983
Earlier work this paper cites.
R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials
1985
Earlier work this paper cites.
M. Rahman, An integral representation of a ϕ 9 10 {}_{10}\phi_{9} and continuous bi-orthogonal ϕ 9 10 {}_{10}\phi_{9} rational functions
1986
Earlier work this paper cites.
R. Askey, Beta integrals in Ramanujan’s papers, his unpublished work and further examples
1988
Earlier work this paper cites.
E. Date, M. Jimbo, A. Kuniba, T. Miwa, and M. Okado, Exactly solvable SOS models, II: Proof of the star-triangle relation and combinatorial identities
1988
Earlier work this paper cites.
J. A. Wilson, Orthogonal functions from Gram determinants
1991
Earlier work this paper cites.
I. M. Gelfand, M. I. Graev, V. S. Retakh, General hypergeometric systems of equations and series of hypergeometric type
1992
Earlier work this paper cites.
L. D. Faddeev, Discrete Heisenberg-Weyl group and modular group
1995
Earlier work this paper cites.
N. Seiberg, Electric–magnetic duality in supersymmetric non-Abelian gauge theories
1995
Earlier work this paper cites.
M. Jimbo and T. Miwa, Quantum KZ equation with | q | = 1 |q|=1 and correlation functions of the XXZ model in the gapless regime
1996
Earlier work this paper cites.
I. B. Frenkel and V. G. Turaev, Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions
1997
Earlier work this paper cites.
S. N. M. Ruijsenaars, First order analytic difference equations and integrable quantum systems
1997
Earlier work this paper cites.
G. E. Andrews, R. Askey, and R. Roy, Special Functions
1999
Earlier work this paper cites.
A. S. Zhedanov, Biorthogonal rational functions and the generalized eigenvalue problem
1999
Earlier work this paper cites.
J. F. van Diejen and V. P. Spiridonov, An elliptic Macdonald-Morris conjecture and multiple modular hypergeometric sums
2000
Earlier work this paper cites.
by same author, Modular double of a quantum group
2000
Earlier work this paper cites.
G. Felder and A. Varchenko, The elliptic gamma function and S L ( 3 , ℤ ) ⋉ ℤ 3 SL(3,\mathbb{Z})\ltimes\mathbb{Z}^{3}
2000
Earlier work this paper cites.
V. P. Spiridonov and A. S. Zhedanov, Spectral transformation chains and some new biorthogonal rational functions
2000
Earlier work this paper cites.
G. E. Andrews, Bailey’s transform, lemma, chains and tree
2001
Earlier work this paper cites.
by same author, Elliptic Selberg integrals
2001
Earlier work this paper cites.
H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations
2001
Earlier work this paper cites.
V. P. Spiridonov, On the elliptic beta function
2001
Earlier work this paper cites.
S. Kharchev, D. Lebedev, and M. Semenov-Tian-Shansky, Unitary representations of U q ( s l ( 2 , ℝ ) ) U_{q}(sl(2,\mathbb{R})) , the modular double and the multiparticle q q -deformed Toda chains
2002
Cited alongside, same era.
by same author, Theta hypergeometric series
2002
Cited alongside, same era.
S. O. Warnaar, Summation and transformation formulas for elliptic hypergeometric series
2002
Cited alongside, same era.
Y. Kajihara and M. Noumi, Multiple elliptic hypergeometric series. An approach from the Cauchy determinant
2003
Cited alongside, same era.
K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, and Y. Yamada, E 9 10 {}_{10}E_{9} solution to the elliptic Painlevé equation
2003
Cited alongside, same era.
by same author, An elementary approach to 6 j 6j -symbols (classical, quantum, rational, trigonometric, and elliptic)
2007
Closest in time.
M. Schlosser, Elliptic enumeration of nonintersecting lattice paths
2007
Closest in time.
by same author, Short proofs of the elliptic beta integrals
2007
Closest in time.
by same author, Elliptic hypergeometric functions and Calogero-Sutherland type models
2007
Closest in time.
by same author, Elliptic grids, rational functions, and the Padé interpolation
2007
Closest in time.
P. J. Forrester and S. O. Warnaar, The importance of the Selberg integral
2008
Closest in time.
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2003
Cited alongside, same era.
by same author, Extensions of the well-poised and elliptic well-poised Bailey lemma
2003
Cited alongside, same era.
E. Friedman and S. Ruijsenaars, Shintani-Barnes zeta and gamma functions
2004
Cited alongside, same era.
G. Gasper and M. Rahman, Basic Hypergeometric Series
2004
Cited alongside, same era.
H. Rosengren, Elliptic hypergeometric series on root systems
2004
Cited alongside, same era.
by same author, Sklyanin invariant integration
2004
Cited alongside, same era.
by same author, Theta hypergeometric integrals
2004
Cited alongside, same era.
by same author, Essays on the theory of elliptic hypergeometric functions
2008
Closest in time.
F. J. van de Bult and E. M. Rains, Basic hypergeometric functions as limits of elliptic hypergeometric functions
2009
Closest in time.
F. A. Dolan and H. Osborn, Applications of the superconformal index for protected operators and q q -hypergeometric identities to 𝒩 = 1 \mathcal{N}=1 dual theories
2009
Closest in time.
A. P. Magnus, Rational interpolation to solutions of Riccati difference equations on elliptic lattices
2009
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by same author, Elliptic hypergeometric solutions to elliptic difference equations
2009
Closest in time.
by same author, Limits of elliptic hypergeometric integrals
2009
Closest in time.
E. M. Rains and V. P. Spiridonov, Determinants of elliptic hypergeometric integrals
2009
Closest in time.
by same author, Continuous biorthogonality of an elliptic hypergeometric function
2009
Closest in time.
by same author, Elliptic polynomials orthogonal on the unit circle with a dense point spectrum
2009
Closest in time.
A. Borodin, V. Gorin, and E. M. Rains, q-Distributions on boxed plane partitions
2010
Closest in time.
A. Gadde, E. Pomoni, L. Rastelli, and S. S. Razamat, S S -duality and 2d topological QFT
2010
Closest in time.
E. M. Rains, Transformations of elliptic hypergeometric integrals
2010
Closest in time.
by same author, An elliptic hypergeometric integral with W ( F 4 ) W(F_{4}) symmetry
2011
Closest in time.
by same author, An isomonodromy interpretation of the hypergeometric solution of the elliptic Painlevé equation (and generalizations)
2011
Closest in time.
by same author, Felder’s elliptic quantum group and elliptic hypergeometric series on the root system A n A_{n}
2011
Closest in time.
by same author, Elliptic hypergeometric terms
2011
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V. P. Spiridonov and G. S. Vartanov, Elliptic hypergeometry of supersymmetric dualities
2011
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by same author, New multiple ψ 6 6 {}_{6}\psi_{6} summation formulas and related conjectures
2011
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W. Zudilin, Arithmetic hypergeometric series
2011
Closest in time.
V. V. Bazhanov and S. M. Sergeev, A master solution of the quantum Yang-Baxter equation and classical discrete integrable equations
2012
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by same author, Elliptic Littlewood identities
2012
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