A Lie theoretic interpretation of multivariate hypergeometric polynomials

TL;DR

This paper provides a Lie algebraic interpretation of multivariate hypergeometric polynomials, revealing their bispectrality and recurrence relations, and offering new proofs of their orthogonality.

Contribution

It introduces a Lie algebra framework for these polynomials, clarifies their duality, and derives new recurrence relations and bispectrality properties.

Findings

Established a Lie algebraic interpretation of the polynomials.

Proved orthogonality using Lie algebra methods.

Derived recurrence relations and demonstrated bispectrality.

Abstract

In 1971 Griffiths used a generating function to define polynomials in d variables orthogonal with respect to the multinomial distribution. The polynomials possess a duality between the discrete variables and the degree indices. In 2004 Mizukawa and Tanaka related these polynomials to character algebras and the Gelfand hypergeometric series. Using this approach they clarified the duality and obtained a new proof of the orthogonality. In the present paper, we interpret these polynomials within the context of the Lie algebra sl_{d+1}. Our approach yields yet another proof of the orthogonality. It also shows that the polynomials satisfy d independent recurrence relations each involving d^2+d+1 terms. This combined with the duality establishes their bispectrality. We illustrate our results with several explicit examples.