Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations

Abstract

An alternative derivation is presented of the infinitely many exceptional Wilson and Askey-Wilson polynomials, which were introduced by the present authors in 2009. Darboux-Crum transformations intertwining the discrete quantum mechanical systems of the original and the exceptional polynomials play an important role. Infinitely many continuous Hahn polynomials are derived in the same manner. The present method provides a simple proof of the shape invariance of these systems as in the corresponding cases of the exceptional Laguerre and Jacobi polynomials.

24 pages. Comments and references added. To appear in J.Phys.A

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• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : III
• Equivalences of the Multi-Indexed Orthogonal Polynomials
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : II
• A new family of shape invariantly deformed Darboux-Pöschl-Teller potentials with continuous \ell
• Exceptional orthogonal polynomials and new exactly solvable potentials in quantum mechanics
• Global Solutions of Certain Second-Order Differential Equations with a High Degree of Apparent Singularity
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials IV : closure relations and creation/annihilation operators
• A short note on "Group theoretic approach to rationally extended shape invariant potentials" [Annals of Physics 359, 46 (2015)]
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• Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q|=1 and Quantum Dilogarithm
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• Exactly and quasi-exactly solvable `discrete' quantum mechanics
• Infinite families of shape invariant potentials with n parameters subject to translation