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arXiv · 1702.03078

New Determinant Expressions of the Multi-indexed Orthogonal Polynomials in Discrete Quantum Mechanics

Satoru Odake
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Abstract

The multi-indexed orthogonal polynomials (the Meixner, little $q$-Jacobi (Laguerre), ($q$-)Racah, Wilson, Askey-Wilson types) satisfying second order difference equations were constructed in discrete quantum mechanics. They are polynomials in the sinusoidal coordinates $\eta(x)$ ($x$ is the coordinate of quantum system) and expressed in terms of the Casorati determinants whose matrix elements are functions of $x$ at various points. By using shape invariance properties, we derive various equivalent determinant expressions, especially those whose matrix elements are functions of the same point $x$. Except for the ($q$-)Racah case, they can be expressed in terms of $\eta$ only, without explicit $x$-dependence.

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Satoru Odake. 2017-02-10. New Determinant Expressions of the Multi-indexed Orthogonal Polynomials in Discrete Quantum Mechanics. https://doi.org/10.1093/ptep/ptx051

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