V. P. Spiridonov
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 integral Bailey chain introduced earlier by the author. Relations to the Sklyanin algebra and elliptic analogues of the Faddeev modular double are discussed in detail.
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We introduce a generalization of elliptic 6j-symbols, which can be interpreted as matrix elements for intertwiners between corepresentations of Felder's elliptic quantum group. For special parameter values, they can be expressed in terms of multivariable elliptic hypergeometric series related to the root system A_n. As a consequence, we obtain new biorthogonality relations for such series.
We derive a number of summation and transformation formulas for elliptic hypergeometric series on the root systems A_n, C_n and D_n. In the special cases of classical and q-series, our approach leads to new elementary proofs of the corresponding identities.
A multiple generalization of elliptic hypergeometric series is investigated and a duality transformation for multiple hypergeometric series is proposed. Our duality transformation obtained from an identity arising from the Cauchy determinant formula for Weierstrass sigma functions, by spcialization of a particular form.
Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system An, with different dimensions n. We give, with a new, elementary, proof, an elliptic analogue of this transformation. We also obtain further Bailey-type transformations as consequences of our result, some of which are new also in the case of basic and classical hypergeometric series.
Using Krattenthaler's operator method, we give a new proof of Warnaar's recent elliptic extension of Krattenthaler's matrix inversion. Further, using a theta function identity closely related to Warnaar's inversion, we derive summation and transformation formulas for elliptic hypergeometric series of Karlsson-Minton-type. A special case yields a particular summation that was used by Warnaar to derive quadratic, cubic and quartic transformations for elliptic hypergeometric series. Starting from another theta function identity, we derive yet different summation and transformation formulas for el…
We use elliptic Taylor series expansions and interpolation to deduce a number of summations for elliptic hypergeometric series. We extend to the well-poised elliptic case results that in the $q$-case have previously been obtained by Cooper and by Ismail and Stanton. We also provide identities involving S. Bhargava's cubic theta functions.
Elliptic Macdonald polynomials of sl(2)-type and level 2 are introduced. Suitable limits of elliptic Macdonald polynomials are the standard Macdonald polynomials and conformal blocks. Identities for elliptic Macdonald polynomials, in particular their modular properties, are studied.
Using multiple q-integrals and a determinant evaluation, we establish a multivariable extension of Bailey's nonterminating 10-phi-9 transformation. From this result, we deduce new multivariable terminating 10-phi-9 transformations, 8-phi-7 summations and other identities. We also use similar methods to derive new multivariable 1-psi-1 summations. Some of our results are extended to the case of elliptic hypergeometric series.
Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating, balanced, very-well-poised, elliptic hypergeometric series.
We consider the space of elliptic hypergeometric functions of the sl_2 type associated with elliptic curves with one marked point. This space represents conformal blocks in the sl_2 WZW model of CFT. The modular group acts on this space. We give formulas for the matrices of the action in terms of values at roots of unity of Macdonald polynomials of the sl_2 type.
We investigate an elliptic quantum group introduced by Felder and Varchenko, which is constructed from the $R$-matrix of the Andrews-Baxter-Forrester model, containing both spectral and dynamical parameter. We explicitly compute the matrix elements of certain corepresentations and obtain orthogonality relations for these elements. Using dynamical representations these orthogonality relations give discrete bi-orthogonality relations for terminating very-well-poised balanced elliptic hypergeometric series, previously obtained by Frenkel and Turaev and by Spiridonov and Zhedanov in different cont…
We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum/integrals associated to the $A_n$ and $BC_n$ root systems, generalising the formulas previously obtained by Rains. The sum/integrals are expressed in terms of the lens elliptic gamma function, a generalisation of the elliptic gamma function that depends on an additional integer variable, as well as a complex variable and two elliptic nomes. As an application of our results, we prove an equality between $S^1\times S^3/\mathbb{Z}_r$ supersymmetric indices, for a pair of four-dimensional $\mathcal{N}=1$ supers…
We propose a new realization of the elliptic quantum group equipped with the H-Hopf algebroid structure on the basis of the elliptic algebra U_{q,p}(\hat{sl}_2). The algebra U_{q,p}(\hat{sl}_2) has a constructive definition in terms of the Drinfeld generators of the quantum affine algebra U_q(\hat{sl}_2) and a Heisenberg algebra. This yields a systematic construction of both finite and infinite-dimensional dynamical representations and their parallel structures to U_q(\hat{sl}_2). In particular we give a classification theorem of the finite-dimensional irreducible pseudo-highest weight represe…
The purpose of this article is to demonstrate that i) the framework of elliptic hypergeometric integrals (EHIs) can be extended by input from supersymmetric gauge theory, and ii) analyzing the hyperbolic limit of the EHIs in the extended framework leads to a rich structure containing sharp mathematical problems of interest to supersymmetric quantum field theorists. Both of the above items have already been discussed in the theoretical physics literature. Item i was demonstrated by Dolan and Osborn in 2008. Item ii was discussed in the present author's Ph.D. Thesis in 2016, wherein crucial elem…
We give an overview of some of the main results from the theories of hypergeometric and basic hypergeometric series and integrals associated with root systems. In particular, we list a number of summations, transformations and explicit evaluations for such multiple series and integrals. We concentrate on such results which do not directly extend to the elliptic level. This text is a provisional version of a chapter on hypergeometric and basic hypergeometric series and integrals associated with root systems for volume 2 of the new Askey--Bateman project which deals with "Multivariable special f…
The trigonometric quantized Knizhnik-Zamolodchikov equation (qKZ equation) associated with the quantum group $U_q(sl_2)$ is a system of linear difference equations with values in a tensor product of $U_q(sl_2)$ Verma modules. We solve the equation in terms of multidimensional $q$-hypergeometric functions and define a natural isomorphism between the space of solutions and the tensor product of the corresponding evaluation Verma modules over the elliptic quantum group $E_{ρ,γ}(sl_2)$, where parameters $ρ$ and $γ$ are related to the parameter $q$ of the quantum group $U_q(sl_2)$ and the step $p$ …
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 function and the Euler gamma function, respectively. The elliptic gamma function appears in Baxter's formula for the free energy of the eight-vertex model and in the hypergeometric solutions of the elliptic qKZB equations. In this paper, the properties of this function are studied. In particular we show that elliptic gamma functions are generalizations of automorphic forms of G=SL(3,Z) x Z^3 associated to a non-…
The representation theory of the Hopf algebroid $U_{q,x}(\widehat{sl_2})=U_{q,λ}(\widehat{sl_2})$ is initiated and it is established that the intertwiner between the tensor products of dynamical evaluation modules is a well-poised balanced $_{10}W_9$ symbol, confirming a conjecture of Konno, that the degeneration of the elliptic $_{12}V_{11}$ series to $_{10}W_9$ can be proven based on the representation theory of $U_{q,x}(\widehat{sl_2})$, the degeneration of the elliptic algebra $U_{q,p}(\widehat{sl_2})$ as $p \to 0$.
A weight-dependent generalization of the binomial theorem for noncommuting variables is presented. This result extends the well-known binomial theorem for q-commuting variables by a generic weight function depending on two integers. For a special case of the weight function, restricting it to depend on only a single integer, the noncommutative binomial theorem involves an expansion of complete symmetric functions. Another special case concerns the weight function to be a suitably chosen elliptic (i.e., doubly-periodic meromorphic) function, in which case an elliptic generalization of the binom…
We consider the Gauss-Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallelly to themselves. We reduce these equations modulo a prime integer $p$ and construct polynomial solutions of the new differential equations as $p$-analogs of the initial hypergeometric integrals. In some cases we interpret the $p$-analogs of the hypergeometric integrals as sums over points of hypersurfaces defined over the finite field $F_p$. That interpretation is similar to the interpretation by Yu.I. Manin in [Ma] of the number of po…
A $τ$ function formalism for Sakai's elliptic Painlev'e equation is presented. This establishes the equivalence between the two formulations by Sakai and by Ohta-Ramani-Grammaticos. We also give a simple geometric description of the elliptic Painlev'e equation as a non-autonomous deformation of the addition formula on elliptic curves. By using these formulations, we construct a particular solution of the elliptic Painlev'e equation expressed in terms of the elliptic hypergeometric function_10E_9.
The purpose of this paper is to introduce and study a q-analogue of the holonomic system of differential equations associated to the Belavin's classical r-matrix (elliptic r-matrix equations), or, equivalently, to define an elliptic deformation of the quantum Knizhnik-Zamolodchikov equations of Frenkel and Reshetikhin. In hep-th 9303018, it was shown that solutions of the elliptic r-matrix equations admit a representation as traces of products of intertwining operators between certain modules over affine sl(N). In this paper, this construction is generalized to quantum affine sl(N). The main o…
By using representation theory of the elliptic quantum group U_{q,p}(sl_N^), we present a systematic method of deriving the weight functions. The resultant sl_N type elliptic weight functions are new and give elliptic and dynamical analogues of those obtained in the trigonometric case. We then discuss some basic properties of the elliptic weight functions. We also present an explicit formula for formal elliptic hypergeometric integral solution to the face type, i.e. dynamical, elliptic q-KZ equation.
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