dorsal/arxiv
View SchemaNon-Symmetric Jack Polynomials and Integral Kernels
| Authors | T. H. Baker, P. J. Forrester |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9612003 |
| URL | https://arxiv.org/abs/q-alg/9612003 |
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 ${\cal N}_\eta$ is evaluated using recurrence relations, and ${\cal N}_\eta$ 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 $\widehat{\Phi}$ and $\widehat{\Psi}$, 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 $\Phi$ introduced recently by Knop and Sahi.
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"abstract": "We investigate some properties of non-symmetric Jack, Hermite and Laguerre\npolynomials which occur as the polynomial part of the eigenfunctions for\ncertain Calogero-Sutherland models with exchange terms. For the non-symmetric\nJack polynomials, the constant term normalization ${\\cal N}_\\eta$ is evaluated\nusing recurrence relations, and ${\\cal N}_\\eta$ is related to the norm for the\nnon-symmetric analogue of the power-sum inner product. Our results for the\nnon-symmetric Hermite and Laguerre polynomials allow the explicit determination\nof the integral kernels which occur in Dunkl\u0027s theory of integral transforms\nbased on reflection groups of type $A$ and $B$, and enable many analogues of\nproperties of the classical Fourier, Laplace and Hankel transforms to be\nderived. The kernels are given as generalized hypergeometric functions based on\nnon-symmetric Jack polynomials. Central to our calculations is the construction\nof operators $\\widehat{\\Phi}$ and $\\widehat{\\Psi}$, which act as lowering-type\noperators for the non-symmetric Jack polynomials of argument $x$ and $x^2$\nrespectively, and are the counterpart to the raising-type operator $\\Phi$\nintroduced recently by Knop and Sahi.",
"arxiv_id": "q-alg/9612003",
"authors": [
"T. H. Baker",
"P. J. Forrester"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Non-Symmetric Jack Polynomials and Integral Kernels",
"url": "https://arxiv.org/abs/q-alg/9612003"
},
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