The multivariate Hahn polynomials and the singular oscillator

TL;DR

This paper introduces multivariate Hahn polynomials derived from the singular oscillator model, detailing their properties, orthogonality, and explicit forms, with extensions to higher dimensions and connections to quantum models.

Contribution

It establishes a novel link between multivariate Hahn polynomials and the singular oscillator, providing explicit properties and generalizations to multiple variables.

Findings

Polynomials depend on d+1 parameters and are orthogonal w.r.t. hypergeometric distribution.

Derived shift operators, recurrence relations, and explicit forms.

Extended results to arbitrary dimensions.

Abstract

Karlin and McGregor's d-variable Hahn polynomials are shown to arise in the (d+1)-dimensional singular oscillator model as the overlap coefficients between bases associated to the separation of variables in Cartesian and hyperspherical coordinates. These polynomials in d discrete variables depend on d+1 real parameters and are orthogonal with respect to the multidimensional hypergeometric distribution. The focus is put on the d=2 case for which the connection with the three-dimensional singular oscillator is used to derive the main properties of the polynomials: forward/backward shift operators, orthogonality relation, generating function, recurrence relations, bispectrality (difference equations) and explicit expression in terms of the univariate Hahn polynomials. The extension of these results to an arbitrary number of variables is presented at the end of the paper.