Nonsymmetric Jack polynomials and integral kernels

Abstract

We investigate some properties of non-symmetric Jack, Hermite and Laguerre polynomials which occur as the polynomial part of the eigenfunctions for certain Calogero-Sutherland models with exchange terms. For the non-symmetric Jack polynomials, the constant term normalization N η is evaluated using recurrence relations, and N η is related to the norm for the non-symmetric analogue of the power-sum inner product. Our results for the non-symmetric Hermite and Laguerre polynomials allow the explicit determination of the integral kernels which occur in Dunkl's theory of integral transforms based on reflection groups of type A and B, and enable many analogues of properties of the classical Fourier, Laplace and Hankel transforms to be derived. The kernels are given as generalized hypergeometric functions based on non-symmetric Jack polynomials. Central to our calculations is the construction of operators Φ and Ψ, which act as lowering-type operators for the non-symmetric Jack polynomials of argument x and x 2 respectively, and are the counterpart to the raising-type operator Φ introduced recently by Knop and Sahi.

Intertwining operators for a degenerate double affine Hecke algebra and multivariable orthogonal polynomials

“…Baker and Forrester named these polynomials non-symmetric Hermite and Laguerre polynomials respectively, and studied their properties [BF2,BF3]. We note that some of their results may be obtained directly form the corresponding properties of the Jack polynomials by applying the intertwiners.…”

“…Wavefunctions for some systems can explicitly be written in terms of suitable special functions. Recent studies on integrable quantum many-particle systems reveal that wavefunctions of some special cases can be written in terms of multivariable analogue of classical orthogonal polynomials [BF1,BF2,BF3,vD,Ka1,Ka2,So,UW]. In the present paper, we shall consider orthogonal polynomials associated with the quantum Calogero models confined in harmonic potential [Ca1,Ca2,Su1,Y]: …”

“…From this viewpoint, each of the models (1.1), (1.2) corresponds to individual representation of the degenerate affine Hecke algebra. So far there have been some works concerning representations of the algebra associated with (rational) Calogero-type models [BF2,BF3,UW]. However intertwiners between representation spaces have not been considered explicitly, though they are important to understand common algebraic structure of the models.…”

Biorthogonal Expansion of Non-Symmetric Jack Functions

“…Here we have used the relations E σ (− 2 y ) = (−2) |σ| E −σ * (y) and y −b E −σ * (y) = E −σ * −b (y); see [2]. Now each term E −σ * −b (y) = E −σ * −b (1 + z) can again be expanded in terms of E κ (z).…”

The product of a nonsymmetric Jack polynomial with a linear function

“…Figure 1. The Young diagram of (7,6,4,3,3,2,1) Presenting the Pieri formula requires some notation. The Young diagram of κ is the set diag(κ) := {(i, j)|1 ≤ j ≤ κ i , 1 ≤ i ≤ n} which is drawn with i increasing from top to bottom and j from left to right.…”