Leonard pairs having LB-TD form

Abstract: Fix an algebraically closed field $\mathbb{F}$ and an integer $d \geq 3$. Let $\text{Mat}{d+1}(\mathbb{F})$ denote the $\mathbb{F}$-algebra consisting of the $(d+1) \times (d+1)$ matrices that have all entries in $\mathbb{F}$. We consider a pair of diagonalizable matrices $A,A^*$ in $\text{Mat}{d+1}(\mathbb{F})$, each acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. Such a pair is called a Leonard pair in $\text{Mat}{d+1}(\mathbb{F})$. 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 $\text{Mat}{d+1}(\mathbb{F})$ 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.