Matrix units associated with the split basis of a Leonard pair

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 → V and A∗ : V → V that satisfy (i) and (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 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.