Leonard pairs and the q-Racah polynomials

Abstract

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider an ordered pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy conditions (i), (ii) below. (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal. We call such a pair a {\it Leonard pair} on $V$. We discuss a correspondence between Leonard pairs and a class of orthogonal polynomials consisting of the $q$-Racah polynomials and some related polynomials of the Askey scheme. For the polynomials in this class we obtain the 3-term recurrence, difference equation, Askey-Wilson duality, and orthogonality in a uniform manner using the corresponding Leonard pair.