Let F denote an algebraically closed field of characteristic 0. Let V denote a vector space over F of finite positive dimension. A Leonard pair on V is an ordered pair of linear transformations in End(V ) such that for each of these transformations there exists a basis for V with respect to which the matrix representing that transformation is diagonal and the matrix representing the other transformation is irreducible tridiagonal. The diameter of the Leonard pair is defined as dim(V )− 1. Let (A, A * ) be a Leonard pair of diameter d. The Leonard pair (A, A * ) is of Krawtchouk type whenever its eigenvalue sequence and its dual eigenvalue sequence are both {d − 2i} d i=0 . The notions of a Leonard triple and a Leonard triple of Krawtchouk type are similarly defined. In this paper, let (A, A * ) be a Leonard pair of Krawtchouk type. We first determine all linear transformations A † in End(V ) such that (A, A * , A † ) is a Leonard triple of Krawtchouk type. Then we classify up to isomorphism Leonard triples of Krawtchouk type.
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