椭圆U（2）量子群与椭圆超几何级数

椭圆超几何函数的连续双正交性

Continuous biorthogonality of the elliptic hypergeometric function

本文构造了一类与高斯超几何函数的椭圆型类似物有关的连续双正交函数族。其中的关键工具是作者在前面介绍的椭圆型Beta积分和积分Bailey链。Sklyanin代数和椭圆类似物的Faddeev模双的关系进行了详细讨论。

椭圆超几何级数在根系An上的新变换

New transformations for elliptic hypergeometric series on the root system An

最近，Kajihara给出了一个Bailey型变换，它将不同维数n的根系An上的基本超几何级数联系起来.我们给，一个新的，基本的，证明，一个椭圆形模拟这个变换。我们还得到进一步的Bailey型变换作为我们的结果的后果，其中一些是新的情况下，也基本和经典的超几何级数。

关于Warnaar椭圆矩阵求逆与Karlsson-Minton型椭圆超几何级数

On Warnaar's elliptic matrix inversion and Karlsson-Minton-type elliptic hypergeometric series

利用Krattenthaler算子方法，给出了Warnaar对Krattenthaler的矩阵求逆的近期椭圆扩张的一个新的证明.进一步，利用与Warnaar反演密切相关的θ函数恒等式，导出了Karlsson-Minton型椭圆超几何级数的求和与变换公式.一个特殊的情况下产生一个特定的总和，被用于Warnaar推导二次，三次和四次变换的椭圆超几何级数。本文从另一个θ函数恒等式出发，导出了Karlsson-Minton型椭圆超几何级数的不同求和与变换公式。这些后者的身份似乎很不寻常，似乎是新的已经在三角（即，p=0）情况下。

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.

多重基本和椭圆超几何级数的行列式求和与变换

Summations and transformations for multiple basic and elliptic hypergeometric series by determinant evaluations

利用多重q-积分和行列式估计，我们建立了Bailey无限10-phi-9变换的多变量扩展。从这个结果出发，我们推出了新的多元终止10-phi-9变换，8-phi-7求和和其它恒等式。我们也使用类似的方法来获得新的多变量1-psi-1求和。我们的一些结果被推广到椭圆超几何级数的情形。

椭圆量子群U_{q,p}（\hat{sl}_2）、Hopf代数体结构和椭圆超几何级数

Elliptic Quantum Group U_{q,p}(\hat{sl}_2), Hopf Algebroid Structure and Elliptic Hypergeometric Series

在椭圆代数U_{q,p}（\hat{sl}_2）的基础上，提出了具有H-Hopf代数体结构的椭圆量子群的一种新实现.利用量子仿射代数U_q（\hat{sl}_2）和Heisenberg代数的Drinfeld生成元，给出了代数U_{q,p}（\hat{sl}_2）的构造性定义.这产生了有限维和无限维动力学表示及其与U_q（\hat{sl}_2）的并行结构的系统构造。特别是，我们给出了一个分类定理的有限维不可约伪最高权表示方面的椭圆模拟的德林费尔德多项式。我们还研究了两个评价表示的张量积的结构，并推导出Clebsch-Gordan系数的椭圆模拟。证明了它可用平衡椭圆超几何级数12 V11表示。

Transformation formulas for bilinear sums of basic hypergeometric series

A master formula of transformation formulas for bilinear sums of basic hypergeometric series is proposed. It is obtained from the author's previous results on a transformation formula for Milne's multivariate generalization of basic hypergeometric series of type $A$ with different dimensions and it can be considered as a generalization of Whipple-Sears transformation formula for terminating balanced ${}_4 ϕ_3$ series. As an application of the master formula, one variable cases of some transformation formulas for bilinear sums of basic hypergeometric series are given as examples. The bilinear t…

基本超几何函数的双边级数

A bilateral series involving basic hypergeometric functions

本文证明了一个双边级数的求和公式，该级数的项是两个基本超几何函数的乘积。在特殊情况下，这类级数作为量子群表示的矩阵元出现。

非交换超几何级数的求和公式

Summation formulae for noncommutative hypergeometric series

建立了含非对易参数和辐角的超几何级数和基本超几何级数的几个求和公式。这些结果受到J. A. Tirao [Proc. Nat. Acad. Sci. 100（14）（2003），8138-8141]。

Hypergeometric and basic hypergeometric series and integrals associated with root systems

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…

Elliptic hypergeometric sum/integral transformations and supersymmetric lens index

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…

多元基本超几何级数的变换公式

Transformation formulae for multivariable basic hypergeometric series

本文研究与（$A$型）Macdonald多项式相关的多元（双边）基本超几何级数.我们得到了这类级数的几个变换和求和性质，包括Heine的${}_2ϕ_1$变换，$q$ -Pfaff-Kummer和欧拉变换，$q$ -Saalschütz求和公式和Sear的终止平衡${}_4ϕ_3$级数的变换.对于双边级数，我们重新推导了${}_1ψ_1$求和公式的Kaneko类比，并给出了Bailey ${}_2ψ_2$变换的多元推广.

基本超几何级数的因式分解

Factorization of Basic Hypergeometric Series

本文研究基本超几何级数的因式分解问题，重点分析一般 $_2ψ_2$ 级数，并建立与根系上基本超几何级数理论的联系，同时给出若干经典求和与变换公式的新证明。