The Rahman polynomials and the Lie algebra $\mathfrak {sl}_3(\mathbb {C})$

Abstract:

We interpret the Rahman polynomials in terms of the Lie algebra $\mathfrak {sl}_3(\mathbb C)$. Using the parameters of the polynomials we define two Cartan subalgebras for $\mathfrak {sl}_3(\mathbb {C})$, denoted $H$ and $\tilde H$. We display an antiautomorphism $\dagger$ of $\mathfrak {sl}_3(\mathbb {C})$ that fixes each element of $H$ and each element of $\tilde H$. We consider a certain finite-dimensional irreducible $\mathfrak {sl}_3(\mathbb {C})$-module $V$ consisting of homogeneous polynomials in three variables. We display a nondegenerate symmetric bilinear form $\langle , \rangle$ on $V$ such that $\langle \beta \xi ,\zeta \rangle = \langle \xi ,\beta ^\dagger \zeta \rangle$ for all $\beta \in \mathfrak {sl}_3(\mathbb {C})$ and $\xi ,\zeta \in V$. We display two bases for $V$; one diagonalizes $H$ and the other diagonalizes $\tilde H$. Both bases are orthogonal with respect to $\langle , \rangle$. We show that when $\langle , \rangle$ is applied to a vector in each basis, the result is a trivial factor times a Rahman polynomial evaluated at an appropriate argument. Thus for both transition matrices between the bases each entry is described by a Rahman polynomial. From these results we recover the previously known orthogonality relation for the Rahman polynomials. We also obtain two seven-term recurrence relations satisfied by the Rahman polynomials, along with the corresponding relations satisfied by the dual polynomials. These recurrence relations show that the Rahman polynomials are bispectral. In our theory the roles of $H$ and $\tilde H$ are interchangable, and for us this explains the duality and bispectrality of the Rahman polynomials. We view the action of $H$ and $\tilde H$ on $V$ as a rank 2 generalization of a Leonard pair.