Casoratian Identities for the Discrete Orthogonal Polynomials in Discrete Quantum Mechanics with Real Shifts

TL;DR

This paper extends Casoratian identities to discrete quantum mechanics with real shifts, providing new identities for q-Racah and related polynomials, building on previous work with other polynomial families.

Contribution

It introduces Casoratian identities for discrete quantum mechanics with real shifts, specifically for q-Racah and its reduced polynomials, expanding the mathematical framework.

Findings

Derived infinitely many Casoratian identities for q-Racah polynomials.

Extended previous identities to the case of real shifts in discrete quantum mechanics.

Connected identities with quantum mechanical formulations of orthogonal polynomials.

Abstract

In our previous papers, the Wronskian identities for the Hermite, Laguerre and Jacobi polynomials and the Casoratian identities for the Askey-Wilson polynomial and its reduced form polynomials were presented. These identities are naturally derived through quantum mechanical formulation of the classical orthogonal polynomials; ordinary quantum mechanics for the former and discrete quantum mechanics with pure imaginary shifts for the latter. In this paper we present the corresponding identities for the discrete quantum mechanics with real shifts. Infinitely many Casoratian identities for the $q$-Racah polynomial and its reduced form polynomials are obtained.