Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 049, 51 pages      arXiv:1103.4593      http://dx.doi.org/10.3842/SIGMA.2012.049

Hermite and Laguerre Symmetric Functions Associated with Operators of Calogero-Moser-Sutherland Type

Patrick Desrosiers a and Martin Hallnäs b
a) Instituto Matemática y Física, Universidad de Talca, 2 Norte 685, Talca, Chile
b) Department of Mathematical Sciences, Loughborough University, Leicestershire, LE11 3TU, UK

Received March 22, 2012, in final form July 25, 2012; Published online August 03, 2012

Abstract
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 so-called 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.

Key words: symmetric functions; super-symmetric polynomials; (deformed) Calogero-Moser-Sutherland models.

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