Orthogonal polynomials and operator orderings

Abstract

An alternative and combinatorial proof is given for a connection between a system of Hahn polynomials and identities for symmetric elements in the Heisenberg algebra, which was first observed by Bender et al. [“Resolution of the operator-ordering problem by the method of finite elements,” Phys. Rev. Lett. 56, 2445 (1986); “Continuous Hahn polynomials and the Heisenberg algebra,” J. Math. Phys. 28, 509 (1987)] and proven by Koornwinder [“Meixner-Pollaczek polynomials and the Heisenberg algebra,” J. Math. Phys. 30, 767 (1989)]. In the same vein two results announced by Bender and Dunne [“Polynomials and operator orderings,” J. Math. Phys. 29, 1727 (1988)] connecting a special one-parameter class of Hermitian operator orderings and the continuous Hahn polynomials are also proven.