1996/12/01 by T. H. Baker, Peter J. Forrester, Baker, T. H. +1 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.q-alg/9612003
openalex publication_date 1996/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Nη is evaluated using recurrence relations, and \cal Nη 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 \widehatΦ and \widehatΨ, which act as lowering-type operators for the non-symmetric Jack polynomials of argument x and x2 respectively, and are the counterpart to the raising-type operator Φ introduced recently by Knop and Sahi.