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

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.