Leonard Pairs Of Bannai/Ito Type Having LB-TD Form

Let IF be an algebraically closed field of characteristic zero and let V be a(d+1)dimensional vector space over IF,where d is a non-negative integer.By a Leonard pair on V,we mean an ordering pair of linear transformations over 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 that of the other transformation is irreducible tridiagonal.By an LB-TD pair on Matd+1(IF),we mean an ordered pair of matrices,such that the first is lower bidiagonal with subdiagonal entries all 1,and the second is irreducible tridiagonal.A Leonard pair(A,A*)on V is said to have LB-TD form whenever there exists a basis for V with respect to which the matrices representing A and A*form an LB-TD pair.In this thesis,we study the Leonard pairs of Bannai/Ito type having LB-TD form.We not only give a sufficient and necessary condition for an LB-TD pair on Matd+1(F)to be a Leonard pair of Bannai/Ito type,but also find all matrix pairs having LB-TD form for Leonard pairs of Bannai/Ito type.The results obtained in this thesis together with those obtained by K.Nomura in[20]and H.Alnajjar in[2]completely solve an open problem to find all Leonard pairs having LB-TD form in Matd+1(IF),which was proposed by P.Terwilliger in[23].