Elliptic Hypergeometric Series Research Articles

Infinite elliptic hypergeometric series: convergence and difference equations

We derive finite difference equations of infinite order for theta-hypergeometric series and investigate the space of their solutions. In general, such infinite series diverge, and we describe some constraints on the parameters when they do converge. In particular, we lift the Hardy-Littlewood criterion of the convergence of $q$-hypergeometric series for ${|q|=1}$, $q^n\neq 1$, to the elliptic level and prove the convergence of infinite very-well poised elliptic hypergeometric $ _{r+1}V_r$-series for restricted values of $q$. Bibliography: 13 titles.

On a generalized Weierstrass theta identity

The classical Weierstrass theta identity plays a central role in the study of modular, elliptic and theta hypergeometric series. In this paper, we establish a multi-parameter Weierstrass theta identity with the use of the Cauchy determinant and an anti-symmetric difference of a product of two theta functions. Some applications of this generalized Weierstrass theta identity to basic and elliptic hypergeometric series are also discussed.

An Elliptic Hypergeometric Function Approach to Branching Rules

We prove Macdonald-type deformations of a number of well-known classical branching rules by employing identities for elliptic hypergeometric integrals and series. We also propose some conjectural branching rules and allied conjectures exhibiting a novel type of vanishing behaviour involving partitions with empty 2-cores.

Non-Stationary Ruijsenaars Functions for κ=t−1/N and Intertwining Operators of Ding-Iohara-Miki Algebra

We construct the non-stationary Ruijsenaars functions (affine analogue of the Macdonald functions) in the special case $\kappa=t^{-1/N}$, using the intertwining operators of the Ding-Iohara-Miki algebra (DIM algebra) associated with $N$-fold Fock tensor spaces. By the $S$-duality of the intertwiners, another expression is obtained for the non-stationary Ruijsenaars functions with $\kappa=t^{-1/N}$, which can be regarded as a natural elliptic lift of the asymptotic Macdonald functions to the multivariate elliptic hypergeometric series. We also investigate some properties of the vertex operator of the DIM algebra appearing in the present algebraic framework; an integral operator which commutes with the elliptic Ruijsenaars operator, and the degeneration of the vertex operators to the Virasoro primary fields in the conformal limit $q \rightarrow 1$.

Multidimensional Matrix Inversions and Elliptic Hypergeometric Series on Root Systems

Multidimensional matrix inversions provide a powerful tool for studying multiple hypergeometric series. In order to extend this technique to elliptic hypergeometric series, we present three new multidimensional matrix inversions. As applications, we obtain a new $A_r$ elliptic Jackson summation, as well as several quadratic, cubic and quartic summation formulas.

Elliptic WP-Bailey Transform and its Applications

In this paper, idea of WP-Bailey transform has been extended to elliptic WP-Bailey transform and it has been applied to establish certain interesting summation and transformation formulas for elliptic and theta hypergeometric series.

Gustafson-Rakha-Type Elliptic Hypergeometric Series

We prove a multivariable elliptic extension of Jackson's summation formula conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson and Rakha. As applications, we obtain two further multivariable elliptic Jackson summations and two multivariable elliptic Bailey transformations. The latter four results are all new even in the trigonometric case.

A $4n$-Point Elliptic Interpolation Formula and its Applications

In this paper, using the technique of the divided difference operators, we introduce a $4n$-point elliptic interpolation formula. Some applications of our interpolation formula to elliptic and basic hypergeometric series are given.

Discrete analogues of Macdonald–Mehta integrals

We consider discretisations of the Macdonald–Mehta integrals from the theory of finite reflection groups. For the classical groups, Ar−1, Br and Dr, we provide closed-form evaluations in those cases for which the Weyl denominators featuring in the summands have exponents 1 and 2. Our proofs for the exponent-1 cases rely on identities for classical group characters, while most of the formulas for the exponent-2 cases are derived from a transformation formula for elliptic hypergeometric series for the root system BCr. As a byproduct of our results, we obtain closed-form product formulas for the (ordinary and signed) enumeration of orthogonal and symplectic tableaux contained in a box.

Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation

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.

Duality Transformation Formulas for Multiple Elliptic Hypergeometric Series of Type BC

New duality transformation formulas are proposed for multiple elliptic hypergeometric series of type BC and of type C. Various transformation and summation formulas are derived as special cases to recover some previously known results.

Felder's Elliptic Quantum Group and Elliptic Hypergeometric Series on the Root System An

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 An. As a consequence, we obtain new biorthogonality relations for such series.

Elliptic quantum group [formula omitted], Hopf algebroid structure and elliptic hypergeometric series

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 ( s l ̂ 2 ) . The algebra U q , p ( s l ̂ 2 ) has a constructive definition in terms of the Drinfeld generators of the quantum affine algebra U q ( s l ̂ 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 ( s l ̂ 2 ) . In particular we give a classification theorem of the finite-dimensional irreducible pseudo-highest weight representations stated in terms of an elliptic analogue of the Drinfeld polynomials. We also investigate a structure of the tensor product of two evaluation representations and derive an elliptic analogue of the Clebsch–Gordan coefficients. We show that it is expressed by using the very-well-poised balanced elliptic hypergeometric series V 11 12 .

ABEL'S METHOD ON SUMMATION BY PARTS FOR ELLIPTIC HYPERGEOMETRIC SERIES

By means of Abel's lemma on summation by parts, a new approach for evaluating elliptic hypergeometric series is presented. Some well-known terminating series identities are reviewed and several new transformation and summation formulae are established.

Theta Functions, Elliptic Hypergeometric Series, and Kawanaka's Macdonald Polynomial Conjecture

We give a new theta-function identity, a special case of which is utilised to prove Kawanaka's Macdonald polynomial conjecture. The theta-function identity further yields a transformation formula for multivariable elliptic hypergeometric series which appears to be new even in the one-variable, basic case.

New transformations for elliptic hypergeometric series on the root system A n

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 extension 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.

On Warnaar's elliptic matrix inversion and Karlsson–Minton-type elliptic 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 elliptic hypergeometric series of Karlsson–Minton type. These latter identities seem quite unusual and appear to be new already in the trigonometric (i.e., p = 0 ) case.

Summation formulae for elliptic hypergeometric series

Several new identities for elliptic hypergeometric series are proved. Remarkably, some of these are elliptic analogues of identities for basic hypergeometric series that are balanced but not very-well-poised.

Sklyanin invariant integration

The Sklyanin algebra admits realizations by difference operators acting on theta functions. Sklyanin found an invariant metric for the action and conjectured an explicit formula for the corresponding reproducing kernel. We prove this conjecture and also give natural biorthogonal and orthogonal bases for the representation space. Moreover, we discuss connections with elliptic hypergeometric series and integrals and with elliptic 6j-symbols.

Elliptic U (2) Quantum Group and Elliptic Hypergeometric Series

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 contexts.