New bispectral polynomials orthogonal on a quadratic bi-lattice are obtained from a truncation of Wilson polynomials. Recurrence relation and difference equation are provided. The recurrence coefficients can be encoded in a perturbed persymmetric Jacobi matrix. The orthogonality relation and an explicit expression in terms of hypergeometric functions are also given. Special cases and connections with the para-Krawtchouk polynomials and the dual-Hahn polynomials are also discussed.

A quantum superintegrable model with reflections on the (n − 1)-sphere is presented. Its symmetry algebra is identified with the higher rank generalization of the Bannai-Ito algebra. It is shown that the Hamiltonian of the system can be constructed from the tensor product of n representations of the superalgebra osp(1|2) and that the superintegrability is naturally understood in that setting. The separated solutions are obtained through the Fischer decomposition and a Cauchy-Kovalevskaia extension theorem.

New analytic spin chains with fractional revival are introduced. Their nearest-neighbor couplings and local magnetic fields correspond to the recurrence coefficients of para-Racah polynomials which are orthogonal on quadratic bi-lattices. These models generalize the spin chain associated to the dual-Hahn polynomials. Instances where perfect state transfer also occurs are identified.

A quantum superintegrable model with reflections on the three-sphere is presented. Its symmetry algebra is identified with the rank-two Bannai-Ito algebra. It is shown that the Hamiltonian of the system can be constructed from the tensor product of four representations of the superalgebra osp(1|2) and that the superintegrability is naturally understood in that setting. The exact separated solutions are obtained through the Fischer decomposition and a Cauchy-Kovalevskaia extension theorem. This paper is dedicated with admiration and gratitude to Jiří Patera and Pavel Winternitz on the occasion of their 80th birthdays.

A supersymmetric extension of the Hahn algebra is introduced. This quadratic superalgebra, which we call the Hahn superalgebra, is constructed using the realization provided by the Dunkl oscillator model in the plane, whose Hamiltonian involves reflection operators. In this realization, the reflections act as grading operators and the odd generators are part of the Schwinger-Dunkl algebra, which is a two-parameter extension of the bosonic su(2) construction. The even part of the algebra is built from bilinears in the odd generators and satisfy the Hahn algebra supplemented with involutions. A family of supersymmetric Dunkl oscillator models in n dimensions is also considered. The Hamiltonians of these supersymmetric models differ from ordinary Dunkl oscillators by pure reflection terms. In two dimensions, the supersymmetric Dunkl oscillator is seen to have the even part of the Hahn superalgebra as invariance algebra.