$BC_n$-symmetric abelian functions

  title={\$BC\_n\$-symmetric abelian functions},
  author={Eric M. Rains},
  journal={Duke Mathematical Journal},
  • E. Rains
  • Published 2004
  • Mathematics
  • Duke Mathematical Journal
We construct a family of BC_n-symmetric biorthogonal abelian functions generalizing Koornwinder's orthogonal polynomials, and prove a number of their properties, most notably analogues of Macdonald's conjectures. The construction is based on a direct construction for a special case generalizing Okounkov's interpolation polynomials. We show that these interpolation functions satisfy a collection of generalized hypergeometric identities, including new multivariate elliptic analogues of… Expand
Q A ] 7 F eb 2 00 4 BC n-symmetric polynomials
We consider two important families of BCn-symmetric polynomials, namely Okounkov's interpolation polynomials and Koornwinder's orthogonal polynomials. We give a family of difference equationsExpand
A nonsymmetric version of Okounkov's BC-type interpolation Macdonald polynomials
Symmetric and nonsymmetric interpolation Laurent polynomials are introduced with the interpolation points depending on $q$ and a $n$-tuple of parameters $\tau=(\tau_1,\ldots,\tau_n)$. For theExpand
Branching rules for symmetric Macdonald polynomials and sl_n basic hypergeometric series
A one-parameter generalisation R_{\lambda}(X;b) of the symmetric Macdonald polynomials and interpolations Macdonald polynomials is studied from the point of view of branching rules. We establish aExpand
An elliptic BCn Bailey Lemma, multiple Rogers-Ramanujan identities and Euler's Pentagonal Number Theorems
An elliptic BC n generalization of the classical two parameter Bailey Lemma is proved, and a basic one parameter BC n Bailey Lemma is obtained as a limiting case. Several summation and transformationExpand
Nonsymmetric interpolation Macdonald polynomials and g_n basic hypergeometric series
The Knop-Sahi interpolation Macdonald polynomials are inhomogeneous and nonsymmetric generalisations of the well-known Macdonald polynomials. In this paper we apply the interpolation MacdonaldExpand
Elliptic Double Affine Hecke Algebras
We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogueExpand
A generalization of Newton's identity and Macdonald functions
  • W. Cai, N. Jing
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. A
  • 2014
A unified method is given to show the existence of Jack and Macdonald polynomials using the generalized Newton identity and a simple proof of the Jing–Jozefiak formula for two-row Macdonald functions is given. Expand
Elliptic Littlewood identities
  • E. Rains
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. A
  • 2012
Ten conjectures for elliptic interpolation functions of [email protected]?s version of the Littlewood identity for (skew) Macdonald polynomials are formulated, each of which can be viewed as a multivariate quadratic transformation, and can be proved in a number of special cases. Expand
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 integralExpand
Multivariate Quadratic Transformations and the Interpolation Kernel
We prove a number of quadratic transformations of elliptic Selberg integrals (conjectured in an earlier paper of the author), as well as studying in depth the ''interpolation kernel'', an analyticExpand


BCn-symmetric polynomials
AbstractWe consider two important families of BCn-symmetric polynomials, namely Okounkov's interpolation polynomials and Koornwinder's orthogonal polynomials. We give a family of difference equationsExpand
Interpolation, Integrality, and a Generalization of Macdonald's Polynomials
In this paper, we introduce a new family of symmetric polynomials which depends on a parameter r. They are defined by specifying certain of their zeros. For the parameter values 1/2, 1, and 2 theyExpand
BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials
We consider 3-parametric polynomialsPμ*(x; q, t, s) which replace theAn-series interpolation Macdonald polynomialsPμ*(x; q, t) for theBCn-type root system. For these polynomials we prove an integralExpand
Spectral Transformation Chains and Some New Biorthogonal Rational Functions
Abstract:A discrete-time chain, associated with the generalized eigenvalue problem for two Jacobi matrices, is derived. Various discrete and continuous symmetries of this integrable equation areExpand
A $q$-beta integral on the unit circle and some biorthogonal rational functions
In this paper we first consider a pair of polynomial sets which are biorthogonal on the unit circle with respect to a complex weight function. We then show how the biorthogonality of this pair ofExpand
Nonsymmetric Koornwinder polynomials and duality
In the fundamental work of Lusztig [L] on affine Hecke algebras, a special role is played by the root system of type Cn. The affine Hecke algebra is a deformation of the group algebra of an affineExpand
Askey-Wilson polynomials for root systems of type BC
This paper introduces a family of Askey-Wilson type orthogonal polynomials in n variables associated with a root system of type BCn. The family depends, apart from q, on 5 parameters. For n = 1 itExpand
Self-dual Koornwinder-Macdonald polynomials
We prove certain duality properties and present recurrence relations for a four-parameter family of self-dual Koornwinder-Macdonald polynomials. The recurrence relations are used to verifyExpand
Elliptic hypergeometric series on root systems
Abstract 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, ourExpand
Symmetric functions and Hall polynomials
I. Symmetric functions II. Hall polynomials III. HallLittlewood symmetric functions IV. The characters of GLn over a finite field V. The Hecke ring of GLn over a finite field VI. Symmetric functionsExpand