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.

arXiv comment29 pages. Comments and references added. To appear in J. Phys. A

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• New Determinant Expressions of the Multi-indexed Orthogonal Polynomials in Discrete Quantum Mechanics
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials V : Racah and $q$-Racah types
• Dual Polynomials of the Multi-Indexed ($q$-)Racah Orthogonal Polynomials
• Type II Multi-indexed Little $q$-Jacobi and Little $q$-Laguerre Polynomials
• Wronskian/Casoratian Identities and their Application to Quantum Mechanical Systems
• Quantum vs Classical Birth and Death Processes; Exactly Solvable Examples
• 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
• Multi-indexed Orthogonal Polynomials of a Discrete Variable and Exactly Solvable Birth and Death Processes
• 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