The classification of Leonard triples that have Bannai/Ito type and odd diameter

Let K denote an algebraically closed field of characteristic zero. Let V denote a vector space over K with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear transformations A, A⁎, Aε in End(V) such that for each B∈{A,A⁎,Aε} there exists a basis for V with respect to which the matrix representing B is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal. The diameter of the Leonard triple (A,A⁎,Aε) is defined to be one less than the dimension of V. In this paper we define a family of Leonard triples said to have Bannai/Ito type. We classify up to isomorphism the Leonard triples that have Bannai/Ito type and odd diameter. We show that each of these Leonard triples satisfies the Z3-symmetric Askey-Wilson relations.