Recurrence Relations of the Multi-Indexed Orthogonal Polynomials

Abstract

Ordinary orthogonal polynomials are uniquely characterized by the three term recurrence relations up to an overall multiplicative constant. We show that the newly discovered M-indexed orthogonal polynomials satisfy 3+2M term recurrence relations with non-trivial initial data of the lowest M+1 members. These include the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. The M=0 case is the corresponding classical orthogonal polynomials.

27 pages. Comments and a reference added, reference information updated. To appear in J.Math.Phys

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• Exactly Solvable Discrete Quantum Mechanical Systems and Multi-indexed Orthogonal Polynomials of the Continuous Hahn and Meixner-Pollaczek Types
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• The Adelic Grassmannian and Exceptional Hermite Polynomials
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• New Determinant Expressions of the Multi-indexed Orthogonal Polynomials in Discrete Quantum Mechanics
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• Dual Polynomials of the Multi-Indexed ($q$-)Racah Orthogonal Polynomials
• On the general family of third-order shape-invariant Hamiltonians related to generalized Hermite polynomials
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types
• Discrete Orthogonality Relations for the Multi-Indexed Orthogonal Polynomials in Discrete Quantum Mechanics with Pure Imaginary Shifts