### Abstract

Consider the Heisenberg Lie algebra with basis X, Y, Z, such that [X, Y] D Z. Then the symmetrization σ(X^{k}Y^{k}) can be written as a polynomial in σ(XY) 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.

Original language | English |
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Pages (from-to) | 41-66 |

Number of pages | 26 |

Journal | Advances in Pure and Applied Mathematics |

Volume | 4 |

Issue number | 1 |

DOIs | |

Publication status | Published - Apr 1 2013 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

**Invariant differential operators on the Heisenberg group and Meixner-Pollaczek polynomials.** / Faraut, Jacques; Wakayama, Masato.

Research output: Contribution to journal › Article

*Advances in Pure and Applied Mathematics*, vol. 4, no. 1, pp. 41-66. https://doi.org/10.1515/apam-2013-0204

}

TY - JOUR

T1 - Invariant differential operators on the Heisenberg group and Meixner-Pollaczek polynomials

AU - Faraut, Jacques

AU - Wakayama, Masato

PY - 2013/4/1

Y1 - 2013/4/1

N2 - Consider the Heisenberg Lie algebra with basis X, Y, Z, such that [X, Y] D Z. Then the symmetrization σ(XkYk) can be written as a polynomial in σ(XY) 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.

AB - Consider the Heisenberg Lie algebra with basis X, Y, Z, such that [X, Y] D Z. Then the symmetrization σ(XkYk) can be written as a polynomial in σ(XY) 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.

UR - http://www.scopus.com/inward/record.url?scp=84878316166&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878316166&partnerID=8YFLogxK

U2 - 10.1515/apam-2013-0204

DO - 10.1515/apam-2013-0204

M3 - Article

VL - 4

SP - 41

EP - 66

JO - Advances in Pure and Applied Mathematics

JF - Advances in Pure and Applied Mathematics

SN - 1867-1152

IS - 1

ER -