Leonard pairs having LB-TD form

Fix an algebraically closed field F and an integer d≥3. Let Matd+1(F) denote the F-algebra consisting of the (d+1)×(d+1) matrices that have all entries in F. We consider a pair of diagonalizable matrices A,A⁎ in Matd+1(F), each acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. Such a pair is called a Leonard pair in Matd+1(F). For a Leonard pair A,A⁎ there is a nonzero scalar q that is used to describe the eigenvalues of A and A⁎. In the present paper we find all Leonard pairs A,A⁎ in Matd+1(F) such that A is lower bidiagonal with subdiagonal entries all 1 and A⁎ is irreducible tridiagonal, under the assumption that q is not a root of unity. This gives a partial solution of a problem given by Paul Terwilliger.