$q$-deformed Griffiths polynomials of Racah type

TL;DR

This paper introduces new $q$-deformed bivariate Griffiths polynomials of Racah type, extending classical orthogonal polynomials with bispectral and biorthogonal properties, and explores their symmetry relations.

Contribution

The authors develop a novel family of $q$-Racah type Griffiths polynomials that generalize previous Racah polynomials and establish their bispectrality and biorthogonality.

Findings

Polynomials are bispectral and biorthogonal.

They generalize classical Griffiths and Racah polynomials.

Symmetry relations are crucial for bispectrality proof.

Abstract

New bivariate Griffiths polynomials of $q$-Racah type are introduced and characterized. They generalize the polynomials orthogonal on the multinomial distribution introduced by R. Griffiths fifty years ago. They also correspond to a $q$-deformation of the Griffiths polynomials of Racah type introduced previously by the authors and collaborators. The latter are recovered from the former by a $q\to1$ limit. We show that these new polynomials are bispectral and biorthogonal. We also exhibit some symmetry relations that are essential in the proof of the bispectrality property.