Multi-indexed Meixner and Little $q$-Jacobi (Laguerre) Polynomials

Abstract

As the fourth stage of the project multi-indexed orthogonal polynomials, we present the multi-indexed Meixner and little $q$-Jacobi (Laguerre) polynomials in the framework of `discrete quantum mechanics' with real shifts defined on the semi-infinite lattice in one dimension. They are obtained, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier, from the quantum mechanical systems corresponding to the original orthogonal polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of virtual state vectors. The virtual state vectors are the solutions of the matrix Schrödinger equation on all the lattice points having negative energies and infinite norm. This is in good contrast to the ($q$-)Racah systems defined on a finite lattice, in which the `virtual state' vectors satisfy the matrix Schrödinger equation except for one of the two boundary points.

29 pages. Comments and references added. To appear in J. Phys. A

References in corpus (5)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• 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
• Dual Christoffel transformations

Cited by in corpus (7)

• Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials
• 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
• Quantum vs Classical Birth and Death Processes; Exactly Solvable Examples
• 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