Darboux transformations from the Appell-Lauricella operator

TL;DR

This paper constructs Darboux transformations for the Appell-Lauricella operator using algebra isomorphisms, leading to quantum integrable systems and polynomial extensions related to the Dirichlet distribution.

Contribution

It introduces a novel algebraic framework linking differential operators and constructs new Darboux transformations for multivariable hypergeometric operators.

Findings

Embedded into commutative algebras of partial differential operators

Constructed quantum completely integrable systems

Extended Jacobi polynomials orthogonal with respect to Dirichlet distribution

Abstract

We define two isomorphic algebras of differential operators: the first algebra consists of ordinary differential operators and contains the hypergeometric differential operator, while the second one consists of partial differential operators in $d$ variables and contains the Appell-Lauricella partial differential operator. Using this isomorphism, we construct partial differential operators which are Darboux transformations from polynomials of the Appell-Lauricella operator. We show that these operators can be embedded into commutative algebras of partial differential operators, containing $d$ mutually commuting and algebraically independent partial differential operators, which can be considered as quantum completely integrable systems. Moreover, these algebras can be simultaneously diagonalized on the space of polynomials leading to extensions of the Jacobi polynomials orthogonal with respect to the Dirichlet distribution on the simplex.