Matrix units associated with the split basis of a Leonard pair

Let denote a field, and let V denote a vector space over 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.