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

## TL;DR

This paper derives new determinant expressions for multi-indexed orthogonal polynomials in discrete quantum mechanics, simplifying their structure and revealing shape invariance properties.

## Contribution

It introduces novel determinant formulas for multi-indexed orthogonal polynomials, emphasizing expressions with matrix elements depending only on the same point, and extends understanding of their shape invariance.

## Key findings

- Derived various equivalent determinant expressions
- Expressed polynomials in terms of sinusoidal coordinates
- Simplified formulas for most cases, excluding ($q$-)Racah

## 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.