Classical Leonard Pairs Having LB-TD Form

Let K denote an algebraically closed field of characteristic zero.LetV denote a nonzero vector space over K with finite dimension.Let End(V)denote the K-algebra consisting of all linear transformations from V to V.By a Leonard pair onV,we mean an ordering pair of linear transformations in End(V)such that for each of these trans-formations there exists a basis forV with respect to which the matrix representing that transformation is diagonal and the matrix representing the other transformation is irre-ducible tridiagonal.Let(A,A~*)be a Leonard pair onV.The Leonard pairs is said to have LB-TD form whenever there exists a basis forV with respect to which the matrix representingis lower bidiagonal with subdiagonal entries all 1 and the matrix representing A~* is irreducible tridiagonal.In this thesis,we discuss the classical Leonard pairs having LB-TD form.We show that the classical Leonard pair(A,A~*)has LB-TD form if and only if(A,A~*)is of classical Racah type or classical Krawtchouk type.