1212.4766 (Ernest G. Kalnins et al.)

Contractions of 2D 2nd order quantum superintegrable systems and the
Askey scheme for hypergeometric orthogonal polynomials    [PDF]

Ernest G. Kalnins, Willard Miller Jr, Sarah Post

We show explicitly that all 2nd order superintegrable systems in 2 dimensions are limiting cases of a single system: the generic 3-parameter potential on the 2-sphere, S9 in our listing. Analogously we show that all of the quadratic symmetry algebras of these systems are contractions of that of S9. By contracting function space realizations of irreducible representations of the S9 algebra (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems we obtain the full Askey scheme of orthogonal hypergeometric polynomials. This relationship directly ties the polynomials and their structure equations to physical phenomena. It is more general because it applies to all special functions that arise from these systems via separation of variables, not just those of hypergeometric type, and it extends to higher dimensions.

View original: http://arxiv.org/abs/1212.4766