$\lambda$-Griffiths polynomials: Bispectrality and biorthogonality

TL;DR

This paper introduces a new family of bivariate polynomials called $\lambda$-Griffiths polynomials, which are bispectral and biorthogonal, with special cases that recover known and new orthogonal polynomial families.

Contribution

It generalizes bivariate Griffiths polynomials by adding a parameter $\lambda$, revealing bispectrality and biorthogonality properties, and identifying parameter values that yield orthogonal polynomials.

Findings

$\lambda$-Griffiths polynomials are bispectral and biorthogonal.

Special parameter values produce orthogonal polynomials.

One special case recovers classical bivariate Griffiths polynomials.

Abstract

We introduce a generalization of bivariate Griffiths polynomials depending on an additional parameter $\lambda$. These $\lambda$-Griffiths polynomials are bivariate, bispectral and biorthogonal. For two specific values of the parameter $\lambda$, they become orthogonal. One of the value is related to the usual bivariate Griffiths polynomials, while the second value produces new orthogonal bivariate polynomials.