Rodrigues Formula for Hi-Jack Symmetric Polynomials Associated with the Quantum Calogero Model

TL;DR

This paper derives a Rodrigues formula for Hi-Jack symmetric polynomials linked to the quantum Calogero model, exploring their properties and relationships with other symmetric polynomials, suggesting they form an orthogonal basis.

Contribution

The paper introduces a Rodrigues formula for Hi-Jack polynomials using Dunkl operators, advancing understanding of their algebraic structure and connections to other polynomial bases.

Findings

Derived Rodrigues formula for Hi-Jack polynomials

Established relationships with Jack polynomials and QISM basis

Proposed Hi-Jack polynomials as orthogonal basis candidates

Abstract

The Hi-Jack symmetric polynomials, which are associated with the simultaneous eigenstates for the first and second conserved operators of the quantum Calogero model, are studied. Using the algebraic properties of the Dunkl operators for the model, we derive the Rodrigues formula for the Hi-Jack symmetric polynomials. Some properties of the Hi-Jack polynomials and the relationships with the Jack symmetric polynomials and with the basis given by the QISM approach are presented. The Hi-Jack symmetric polynomials are strong candidates for the orthogonal basis of the quantum Calogero model.