Multi-indexed (q-)Racah Polynomials

Abstract

As the second stage of the project multi-indexed orthogonal polynomials, we present, in the framework of `discrete quantum mechanics' with real shifts in one dimension, the multi-indexed (q-)Racah polynomials. They are obtained from the (q-)Racah polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of `virtual state' vectors, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier. The virtual state vectors are the `solutions' of the matrix Schrödinger equation with negative `eigenvalues', except for one of the two boundary points.

29 pages. The type II (q-)Racah polynomials are deleted because they can be obtained from the type I polynomials. To appear in J.Phys.A

References in corpus (15)

• Mirror Inversion of Quantum States in Linear Registers
• Infinitely many shape invariant potentials and new orthogonal polynomials
• Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials
• Two-step Darboux transformations and exceptional Laguerre polynomials
• Solvable rational extensions of the isotonic oscillator
• Orthogonal Polynomials from Hermitian Matrices
• Higher-order SUSY, exactly solvable potentials, and exceptional orthogonal polynomials
• 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
• Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials
• Rationally-extended radial oscillators and Laguerre exceptional orthogonal polynomials in kth-order SUSYQM
• The Exceptional (X_{\ell}) (q)-Racah Polynomials
• Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations
• Prepotential approach to solvable rational potentials and exceptional orthogonal polynomials
• Dual Christoffel transformations

Cited by in corpus (29)

• Extensions of solvable potentials with finitely many discrete eigenstates
• Orthogonal Polynomials from Hermitian Matrices II
• A new recurrence formula for generic exceptional orthogonal polynomials
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : III
• Casoratian Identities for the Wilson and Askey-Wilson Polynomials
• DARBOUX partners of pseudoscalar Dirac potentials associated with exceptional orthogonal polynomials
• Multi-indexed Meixner and Little $q$-Jacobi (Laguerre) Polynomials
• 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 Discrete Quantum Mechanical Systems and Multi-indexed Orthogonal Polynomials of the Continuous Hahn and Meixner-Pollaczek Types
• Exactly solvable discrete time Birth and Death processes
• Perturbations around the zeros of classical orthogonal polynomials
• New Determinant Expressions of the Multi-indexed Orthogonal Polynomials in Discrete Quantum Mechanics
• Discrete orthogonality relations for multi-indexed Laguerre and Jacobi polynomials
• Non-polynomial extensions of solvable potentials a la Abraham-Moses
• Wronskian/Casoratian Identities and their Application to Quantum Mechanical Systems
• 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
• 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
• Quantum vs Classical Birth and Death Processes; Exactly Solvable Examples
• 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