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Advances in Pure and Applied Mathematics

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Invariant differential operators on the Heisenberg group and Meixner–Pollaczek polynomials

Jacques Faraut
  • Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4 place Jussieu, case 247, 75 252 Paris cedex 05, France
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/ Masato Wakayama
Published Online: 2013-04-01 | DOI: https://doi.org/10.1515/apam-2013-0204


Consider the Heisenberg Lie algebra with basis

, such that
. Then the symmetrization
can be written as a polynomial in
and Z, and this polynomial is identified as a Meixner–Pollaczek polynomial. This is an observation by Bender, Mead and Pinsky, a proof of which has been given by Koornwinder. We extend this result in the framework of Gelfand pairs associated with the Heisenberg group. This extension involves multivariable Meixner–Pollaczek polynomials.

Keywords: Heisenberg group; Gelfand pair; spherical function; Laguerre polynomial; Meixner–Pollaczek polynomial

About the article

Received: 2012-10-29

Accepted: 2013-01-10

Published Online: 2013-04-01

Published in Print: 2013-04-01

Citation Information: Advances in Pure and Applied Mathematics, Volume 4, Issue 1, Pages 41–66, ISSN (Online) 1869-6090, ISSN (Print) 1867-1152, DOI: https://doi.org/10.1515/apam-2013-0204.

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