ar X iv : q - a lg / 9 61 20 03 v 1 1 D ec 1 99 6 Non – Symmetric Jack Polynomials and Integral Kernels
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چکیده
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.
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تاریخ انتشار 1996