Casoratian Identities for the Wilson and Askey-Wilson Polynomials

Abstract

Infinitely many Casoratian identities are derived for the Wilson and Askey-Wilson polynomials in parallel to the Wronskian identities for the Hermite, Laguerre and Jacobi polynomials, which were reported recently by the present authors. These identities form the basis of the equivalence between eigenstate adding and deleting Darboux transformations for solvable (discrete) quantum mechanical systems. Similar identities hold for various reduced form polynomials of the Wilson and Askey-Wilson polynomials, e.g. the continuous q-Jacobi, continuous (dual) (q-)Hahn, Meixner-Pollaczek, Al-Salam-Chihara, continuous (big) q-Hermite, etc.

31 pages, 2 figures. Comments and references added. To appear in Journal of Approximation Theory

References in corpus (7)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• Orthogonal Polynomials from Hermitian Matrices
• Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states
• Exact solution in the Heisenberg picture and annihilation-creation operators
• Orthogonal Polynomials from Hermitian Matrices II
• q-oscillator from the q-Hermite Polynomial

Cited by in corpus (13)

• 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
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials IV : closure relations and creation/annihilation operators
• Exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice
• 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
• Reflectionless Potentials for Difference Schrödinger Equations
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials V : Racah and $q$-Racah types
• New Finite Type Multi-Indexed Orthogonal Polynomials Obtained From State-Adding Darboux Transformations
• Casoratian Identities for the Discrete Orthogonal Polynomials in Discrete Quantum Mechanics with Real Shifts
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types