Dual Christoffel transformations

Abstract

Crum's theorem and its modification a la Krein-Adler are formulated for the discrete quantum mechanics with real shifts, whose eigenfunctions consist of orthogonal polynomials of a discrete variable. The modification produces the associated polynomials with a finite number of degrees deleted. This in turn provides the well known Christoffel transformation for the dual orthogonal polynomials with the corresponding positions deleted.

41 pages, 4 figures. Comments added. To appear in Prog. Theor. Phys

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• Exactly Solvable Discrete Quantum Mechanical Systems and Multi-indexed Orthogonal Polynomials of the Continuous Hahn and Meixner-Pollaczek Types
• 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
• Type II Multi-indexed Little $q$-Jacobi and Little $q$-Laguerre Polynomials
• Wronskian/Casoratian Identities and their Application to Quantum Mechanical Systems
• Casoratian Identities for the Discrete Orthogonal Polynomials in Discrete Quantum Mechanics with Real Shifts
• "Diophantine'' and Factorisation Properties of Finite Orthogonal Polynomials in the Askey Scheme
• New Finite Type Multi-Indexed Orthogonal Polynomials Obtained From State-Adding Darboux Transformations
• Multi-indexed Orthogonal Polynomials of a Discrete Variable and Exactly Solvable Birth and Death Processes
• Discrete Orthogonality Relations for the Multi-Indexed Orthogonal Polynomials in Discrete Quantum Mechanics with Pure Imaginary Shifts
• Some Difference Relations for Orthogonal Polynomials of a Continuous Variable in the Askey Scheme
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types