Jordan algebras and orthogonal polynomials

Abstract

We illustrate how Jordan algebras can provide a framework for the interpretation of certain classes of orthogonal polynomials. The big -1 Jacobi polynomials are eigenfunctions of a first order operator of Dunkl type. We consider an algebra that has this operator (up to constants) as one of its three generators and whose defining relations are given in terms of anticommutators. It is a special case of the Askey-Wilson algebra AW(3). We show how the structure and recurrence relations of the big -1 Jacobi polynomials are obtained from the representations of this algebra. We also present ladder operators for these polynomials and point out that the big -1 Jacobi polynomials satisfy the Hahn property with respect to a generalized Dunkl operator.

11 pages, 30 references

References in corpus (3)

• A Recurrence Relation Approach to Higher Order Quantum Superintegrability
• Supersymmetric Quantum Mechanics with Reflections
• A Bochner Theorem for Dunkl Polynomials

Cited by in corpus (6)

• The Bannai-Ito algebra and some applications
• Cat-States in the Framework of Wigner-Heisenberg Algebra
• The Bannai-Ito algebra and a superintegrable system with reflections on the 2-sphere
• The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)
• An infinite family of superintegrable Hamiltonians with reflection in the plane
• Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q|=1 and Quantum Dilogarithm