Dual Polynomials of the Multi-Indexed ($q$-)Racah Orthogonal Polynomials

TL;DR

This paper studies dual polynomials of multi-indexed ($q$-)Racah orthogonal polynomials, revealing their properties as ordinary orthogonal polynomials and their application in exactly solvable discrete quantum mechanics.

Contribution

It introduces dual multi-indexed ($q$-)Racah polynomials, showing they satisfy three-term recurrence relations and higher-order difference equations, and constructs new quantum systems based on them.

Findings

Dual polynomials satisfy three-term recurrence relations.

Dual polynomials are ordinary orthogonal and Krall-type.

New exactly solvable quantum models are constructed.

Abstract

We consider dual polynomials of the multi-indexed ($q$-)Racah orthogonal polynomials. The $M$-indexed ($q$-)Racah polynomials satisfy the second order difference equations and various $1+2L$ ($L\geq M+1$) term recurrence relations with constant coefficients. Therefore their dual polynomials satisfy the three term recurrence relations and various $2L$-th order difference equations. This means that the dual multi-indexed ($q$-)Racah polynomials are ordinary orthogonal polynomials and the Krall-type. We obtain new exactly solvable discrete quantum mechanics with real shifts, whose eigenvectors are described by the dual multi-indexed ($q$-)Racah polynomials. These quantum systems satisfy the closure relations, from which the creation/annihilation operators are obtained, but they are not shape invariant.