The Classification of Leonard Triples of Bannai/Ito Type with Even Diameters

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*, A ^? in End(V) such that for each B ∈ {A, A*, A ^?} 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*, A ^?) is defined to be one less than the dimension of V. In this paper we define a family of Leonard triples said to be Bannai/Ito type and classify these Leonard triples with even diameters up to isomorphism. Moreover, we show that each of them satisfies the ?₃-symmetric Askey–Wilson relations.