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.
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