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

Jacques Faraut; Masato Wakayama

doi:10.1515/apam-2013-0204

Published by De Gruyter April 1, 2013

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

Jacques Faraut and Masato Wakayama

From the journal Advances in Pure and Applied Mathematics

https://doi.org/10.1515/apam-2013-0204

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Abstract.

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

Received: 2012-10-29

Accepted: 2013-01-10

Published Online: 2013-04-01

Published in Print: 2013-04-01

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

Abstract.

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