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

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.

43 pages. Typos corrected, reference numbering changed, journal data updated. To appear in PTEP

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Cited by in corpus (3)

• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials V : Racah and $q$-Racah types
• Discrete Orthogonality Relations for the Multi-Indexed Orthogonal Polynomials in Discrete Quantum Mechanics with Pure Imaginary Shifts
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types