The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)

TL;DR

This paper derives the algebra of dual -1 Hahn polynomials and demonstrates its role in the Clebsch-Gordan problem of the algebra sl_{-1}(2), revealing new algebraic structures and representations related to quantum and parabose oscillators.

Contribution

It introduces the algebra H from dual -1 Hahn polynomials and links it to the Clebsch-Gordan problem of sl_{-1}(2), including new realizations and representations.

Findings

H algebra derived from dual -1 Hahn polynomials

H can be realized via two sl_{-1}(2) algebras

Clebsch-Gordan coefficients are dual -1 Hahn polynomials

Abstract

The algebra H of the dual -1 Hahn polynomials is derived and shown to arise in the Clebsch-Gordan problem of sl_{-1}(2). The dual -1 Hahn polynomials are the bispectral polynomials of a discrete argument obtained from a q-> -1 limit of the dual q-Hahn polynomials. The Hopf algebra sl_{-1}(2) has four generators including an involution, it is also a q-> -1 limit of the quantum algebra sl_{q}(2) and furthermore, the dynamical algebra of the parabose oscillator. The algebra H, a two-parameter generalization of u(2) with an involution as additional generator, is first derived from the recurrence relation of the -1 Hahn polynomials. It is then shown that H can be realized in terms of the generators of two added sl_{-1}(2) algebras, so that the Clebsch-Gordan coefficients of sl_{-1}(2) are dual -1 Hahn polynomials. An irreducible representation of H involving five-diagonal matrices and connected to the difference equation of the dual -1 Hahn polynomials is constructed.