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Hermite and Laguerre symmetric functions associated with operators of Calogero-Moser-Sutherland type

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posted on 28.02.2013, 15:18 authored by Patrick Desrosiers, Martin Hallnas
We introduce and study natural generalisations of the Hermite and Laguerre polynomials in the ring of symmetric functions as eigenfunctions of infinite-dimensional analogues of partial differential operators of Calogero-Moser-Sutherland (CMS) type. In particular, we obtain generating functions, duality relations, limit transitions from Jacobi symmetric functions, and Pieri formulae, as well as the integrability of the corresponding operators. We also determine all ideals in the ring of symmetric functions that are spanned by either Hermite or Laguerre symmetric functions, and by restriction of the corresponding infinite-dimensional CMS operators onto quotient rings given by such ideals we obtain socalled deformed CMS operators. As a consequence of this restriction procedure, we deduce, in particular, infinite sets of polynomial eigenfunctions, which we shall refer to as super Hermite and super Laguerre polynomials, as well as the integrability, of these deformed CMS operators. We also introduce and study series of a generalised hypergeometric type, in the context of both symmetric functions and 'super' polynomials.



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DESROSIERS, P. and HALLNÄS, M., 2012. Hermite and Laguerre symmetric functions associated with operators of Calogero-Moser-Sutherland type. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 8 (049), 51pp.


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This article was published in the journal, Symmetry, Integrability and Geometry : Methods and Applications (SIGMA) [© the authors] and is made available under published in SIGMA under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence: http://creativecommons.org/licenses/by-nc-sa/3.0/