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

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 the 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. (C) 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4790417]