Orthogonal polynomials and operator orderings

TL;DR

This paper provides a combinatorial proof linking Hahn and continuous Hahn polynomials to symmetric elements in the Heisenberg algebra, expanding understanding of operator orderings and their polynomial connections.

Contribution

It offers a new combinatorial proof of known polynomial identities and extends results relating Hermitian operator orderings to Hahn polynomials.

Findings

Established a combinatorial proof for Hahn polynomial identities

Connected Hermitian operator orderings with continuous Hahn polynomials

Extended previous results on operator polynomial relations

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, Mead, and Pinsky [Phys. Rev. Lett. 56 (1986), J. Math. Phys. 28, 509 (1987)] and proved by Koornwinder [J. Phys. Phys. 30(4), 1989]. In the same vein two results announced by Bender and Dunne [J. Math. Phys. 29 (8), 1988] connecting a special one-parameter class of Hermitian operator orderings and the continuous Hahn polynomials are also proved.