nonzero entries only on the main diagonal, on the superdiagonal and the subdiagonal. A tridiagonal matrix is called irreducible whenever all entries on the superdiagonal and subdiagonal are nonzero.

\[deflp\] Let $V$ be a vector space over ${{\Bbb K}}$ with finite positive dimension. By a [*Leonard pair*]{} on $V$ we mean an ordered pair $(A,A^*)$, where $A:V\to
V$ and $A^*:V\to V$ are linear transformations which satisfy the following two conditions:

1.  There exists a basis for $V$ with respect to which the matrix representing $A^*$ is diagonal, and the matrix representing $A$ is irreducible tridiagonal.

2.  There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal, and the matrix representing $A^*$ is irreducible tridiagonal.

In this paper we do not use the conventional notation $A^*$ for the conjugate-transpose of $A$. In a Leonard pair $(A,A^*)$, the linear transformations $A$ and $A^*$ are arbitrary subject to the conditions [*(i)*]{} and [*(ii)*]{} above.

Leonard pairs occur in the theory of orthogonal polynomials, combinatorics, the representation theory of the Lie algebra $sl_2$ or the quantum group $U_q(sl_2)$. We refer to [@terwgen] as a survey on Leonard pairs, and as a source of further references.

We have the 