A Lie-theoretic interpretation of multivariate hypergeometric polynomials
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, \textit{H. Mizukawa} and \textit{H. Tanaka} [Proc. Am. Math. Soc. 132, No. 9, 2613--2618 (2004; Zbl 1059.33020)] 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.NEWLINENEWLINEIn the present paper, these polynomials are interpreted within the context of the Lie algebra \(sl_{d+1}(\mathbb{C})\). This approach yields yet another proof of the orthogonality. Moreover, it shows that the polynomials satisfy \(d\) independent recurrence relations each involving \(d^2+d+1\) terms. This, combined with the duality, establishes their bispectrality. Finally, the results are illustrated with several explicit examples.
scientific article; zbMATH DE number 6051034