  Let $K$ denote a field, and let $V$ denote a vector space over $K$ with
finite positive dimension. We consider a pair of linear transformations $A:V
\to V$ and $A^*:V \to V$ that satisfy (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 irreducible tridiagonal and the matrix representing $A$
is diagonal.
  We call such a pair a {\em Leonard pair} on $V$. It is known that there
exists a basis for $V$ with respect to which the matrix representing $A$ is
lower bidiagonal and the matrix representing $A^*$ is upper bidiagonal. In this
paper we give some formulae involving the matrix units associated with this
basis.
