Hahn polynomials on polyhedra and quantum integrability

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.