Hahn polynomials on polyhedra and quantum integrability

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.

References in corpus (3)

• A higher rank Racah algebra and the ${\mathbb{Z}}_2^n$ Laplace-Dunkl operator
• Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere
• The generic quantum superintegrable system on the sphere and Racah operators

Cited by in corpus (7)

• Representations of the rank two Racah algebra and orthogonal multivariate polynomials
• Racah problems for the oscillator algebra, the Lie algebra ${\mathfrak{sl}}_n$, and multivariate Krawtchouk polynomials
• The $q$-Bannai-Ito algebra and multivariate $(-q)$-Racah and Bannai-Ito polynomials
• Hahn polynomials for hypergeometric distribution
• Gaudin model for the multinomial distribution
• Spectral analysis of an open $q$-difference Toda chain with two-sided boundary interactions on the finite integer lattice
• A discrete realization of the higher rank Racah algebra