Hahn polynomials on polyhedra and quantum integrability

TL;DR

This paper introduces orthogonal Hahn polynomials on polyhedral domains, explores their connection to quantum superintegrable systems, and demonstrates their algebraic and symmetry properties, extending continuous models to discrete settings.

Contribution

It explicitly constructs Hahn polynomials on polyhedra, characterizes them as eigenfunctions of commuting operators, and links them to quantum integrability and symmetry algebras.

Findings

Hahn polynomials are explicitly defined on polyhedral domains.

They serve as eigenfunctions of commuting difference operators.

The discrete system mirrors properties of continuous quantum superintegrable systems.

Abstract

Orthogonal polynomials with respect to the hypergeometric distribution on lattices in polyhedral domains in ${\mathbb R}^d$, which include hexagons in ${\mathbb R}^2$ and truncated tetrahedrons in ${\mathbb R}^3$, are defined and studied. The polynomials are given explicitly in terms of the classical one-dimensional Hahn polynomials. They are also characterized as common eigenfunctions of a family of commuting partial difference operators. These operators provide symmetries for a system that can be regarded as a discrete extension of the generic quantum superintegrable system on the $d$-sphere. Moreover, the discrete system is proved to possess all essential properties of the continuous system. In particular, the symmetry operators for the discrete Hamiltonian define a representation of the Kohno-Drinfeld Lie algebra on the space of orthogonal polynomials, and an explicit set of $2d-1$ generators for the symmetry algebra is constructed. Furthermore, other discrete quantum superintegrable systems, which extend the quantum harmonic oscillator, are obtained by considering appropriate limits of the parameters.