TL;DR
This paper introduces multi-indexed (q-)Racah polynomials derived via Darboux transformations in discrete quantum mechanics, expanding the class of multi-indexed orthogonal polynomials with potential applications in mathematical physics.
Contribution
It presents the construction of multi-indexed (q-)Racah polynomials using discrete Darboux transformations, extending previous work on Laguerre and Jacobi polynomials.
Findings
Construction method for multi-indexed (q-)Racah polynomials
Extension of multi-indexed orthogonal polynomial framework
Application of virtual state vectors in polynomial generation
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 Schr\"odinger equation with negative `eigenvalues', except for one of the two boundary points.
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