The Classification Of Leonard Triples That Have Bannai/Ito Type And Even Diameters

Let IK denote an algebraically closed field of characteristic zero. Let V denote a vector space over K with finite positive dimension. Let A:Vâ†'V, A*:Vâ†'V and Aε:Vâ†'V denote linear transformations on V. 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 thesis, we discuss the classification of Leonard triples that have Bannai/Ito type with even diameter. This thesis is divided into the following there parts:In Part1, we recall some basic concepts and results concerning Leonard pairs and Leonard systems.In Part2, we define a family of Leonard pairs called Bannai/Ito type and classify up to isomorphism these Leonard triples with even diameters. Moreover, we give the corresponding Askey-Wilson relations and the Z3-symmetric Askey-Wilson relations.In Part3, we define a family of Leonard triples called Bannai/Ito type and we classify up to isomorphism these Leonard triples with even diameters. Moreover, we show that each of them satisfies the Z3-symmetric Askey-Wilson relations.