2012/12/31 by E. G. Kalnins, Ernest G. Kalnins, Willard Miller Jr +1 · 77 citations
Mathematics · Physics and Astronomy · #Algebra over a field #Hypergeometric distribution #Hypergeometric function #Mathematics #Nonlinear Waves and Solitons #Orthogonal polynomials #Pure mathematics #Quadratic equation #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #math-ph #math.MP #nlin.SI
paper · pdf · doi:10.3842/sigma.2013.057
published in Symmetry Integrability and Geometry Methods and Applications (National Academy of Sciences of Ukraine)
arxiv created 2013/10/02 · openalex publication_date 2013/10/02 · arxiv updated 2013/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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. We extend the Wigner-Inn method of Lie algebra contractions to contractions of quadratic algebras and show that all of the quadratic symmetry algebras of these systems are contractions of that of S9. Amazingly, all of the relevant contractions of these superintegrable systems on flat space and the sphere are uniquely induced by the well known Lie algebra contractions of e(2) and so(3). 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, and using Wigner's idea of "saving" a representation, we obtain the full Askey scheme of hypergeometric orthogonal 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.