Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : III

Abstract

In a previous paper, we presented conjectures of the recurrence relations with constant coefficients for the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. In this paper we present a proof for the Laguerre and Jacobi cases. Their bispectral properties are also discussed, which give a method to obtain the coefficients of the recurrence relations explicitly. This paper extends to the Laguerre and Jacobi cases the bispectral techniques recently introduced by Gómez-Ullate et al. to derive explicit expressions for the coefficients of the recurrence relations satisfied by exceptional polynomials of Hermite type.

37 pages. Comments added, typo in (A.15) corrected, reference information updated. To appear in J.Math.Phys

References in corpus (8)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• Exceptional orthogonal polynomials and the Darboux transformation
• Novel Enlarged Shape Invariance Property and Exactly Solvable Rational Extensions of the Rosen-Morse II and Eckart Potentials
• Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations
• A new recurrence formula for generic exceptional orthogonal polynomials
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : II
• Global Solutions of Certain Second-Order Differential Equations with a High Degree of Apparent Singularity

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• Exactly Solvable Discrete Quantum Mechanical Systems and Multi-indexed Orthogonal Polynomials of the Continuous Hahn and Meixner-Pollaczek Types
• Quantum models with energy-dependent potentials solvable in terms of exceptional orthogonal polynomials
• Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials
• New Determinant Expressions of the Multi-indexed Orthogonal Polynomials in Discrete Quantum Mechanics
• Dual Polynomials of the Multi-Indexed ($q$-)Racah Orthogonal Polynomials
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials V : Racah and $q$-Racah types
• Wronskian/Casoratian Identities and their Application to Quantum Mechanical Systems
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types