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SIGMA, 2012, Volume 8, 049, 51 pages (Mi sigma726)  

This article is cited in 6 scientific papers (total in 6 papers)

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

Patrick Desrosiersa, Martin Hallnäsb

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

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.

Keywords: symmetric functions, super-symmetric polynomials, (deformed) Calogero–Moser–Sutherland models.


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ArXiv: 1103.4593
MSC: 05E05; 13J05; 81R12
Received: March 22, 2012; in final form July 25, 2012; Published online August 3, 2012

Citation: Patrick Desrosiers, Martin Hallnäs, “Hermite and Laguerre symmetric functions associated with operators of Calogero–Moser–Sutherland type”, SIGMA, 8 (2012), 049, 51 pp.

Citation in format AMSBIB
\by Patrick Desrosiers, Martin Halln\"as
\paper Hermite and Laguerre symmetric functions associated with operators of Calogero--Moser--Sutherland type
\jour SIGMA
\yr 2012
\vol 8
\papernumber 049
\totalpages 51

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    This publication is cited in the following articles:
    1. Luc Lapointe, Pierre Mathieu, “From Jack to Double Jack Polynomials via the Supersymmetric Bridge”, SIGMA, 11 (2015), 051, 15 pp.  mathnet  crossref  mathscinet
    2. Desrosiers P., Liu D.-Zh., “Selberg Integrals, Super-Hypergeometric Functions and Applications to Beta-Ensembles of Random Matrices”, Random Matrices-Theor. Appl., 4:2 (2015), 1550007  crossref  mathscinet  zmath  isi
    3. G. I. Olshanski, “Diffusion processes on the Thoma cone”, Funct. Anal. Appl., 50:3 (2016), 237–240  mathnet  crossref  crossref  mathscinet  isi  elib
    4. Olshanski G., “The representation ring of the unitary groups and Markov processes of algebraic origin”, Adv. Math., 300 (2016), 544–615  crossref  mathscinet  zmath  isi  scopus
    5. Felipe van Diejen J., Emsiz E., “Branching Rules For Symmetric Hypergeometric Polynomials”, Representation Theory, Special Functions and Painleve Equations - Rims 2015, Advanced Studies in Pure Mathematics, 76, eds. Konno H., Sakai H., Shiraishi J., Suzuki T., Yamada Y., Math Soc Japan, 2018, 125–153  mathscinet  isi
    6. Atai F., Hallnas M., Langmann E., “Orthogonality of Super-Jack Polynomials and a Hilbert Space Interpretation of Deformed Calogero-Moser-Sutherland Operators”, Bull. London Math. Soc., 51:2 (2019), 353–370  crossref  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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