Equivalences of the Multi-Indexed Orthogonal Polynomials

TL;DR

This paper investigates the equivalences among multi-indexed orthogonal polynomials associated with shape-invariant quantum systems, revealing how different index sets can produce proportional polynomials with shifted parameters.

Contribution

It clarifies the conditions under which multi-indexed orthogonal polynomials of various types are equivalent, extending understanding of their structure and parameter relations.

Findings

Different index sets can produce equivalent polynomials.

Proportionality between polynomials with mixed indices and shifted parameters.

Equivalence relations are established for Laguerre, Jacobi, Wilson, and Askey-Wilson types.

Abstract

Multi-indexed orthogonal polynomials describe eigenfunctions of exactly solvable shape-invariant quantum mechanical systems in one dimension obtained by the method of virtual states deletion. Multi-indexed orthogonal polynomials are labeled by a set of degrees of polynomial parts of virtual state wavefunctions. For multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types, two different index sets may give equivalent multi-indexed orthogonal polynomials. We clarify these equivalences. Multi-indexed orthogonal polynomials with both type I and II indices are proportional to those of type I indices only (or type II indices only) with shifted parameters.