2011/03/31 by Patrick Desrosiers, Martin Hallnäs · 1 citation
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Classical orthogonal polynomials #Differential operator #Eigenfunction #Eigenvalues and eigenvectors #Hermite polynomials #Hypergeometric function #Laguerre polynomials #Mathematics #Nonlinear Waves and Solitons #Orthogonal polynomials #Physics #Pure mathematics #Ring of symmetric functions #Symmetric function #Type (biology) #math-ph #math.MP #math.QA
paper · pdf · doi:10.3842/sigma.2012.049
published as SIGMA 8 (2012), 049, 51 pages
arxiv created 2012/08/03 · openalex publication_date 2012/08/03 · arxiv updated 2012/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.