Leonard pairs having specified end-entries

Fix an algebraically closed field $\mathbb{F}$ and an integer $d \geq 3$. Let $V$ be a vector space over $\mathbb{F}$ with dimension $d+1$. A Leonard pair on $V$ is an ordered pair of diagonalizable linear transformations $A: V \to V$ and $A^* : V \to V$, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. Let $\{v_i\}_{i=0}^d$ (resp.\ $\{v^*_i\}_{i=0}^d$) be such an eigenbasis for $A$ (resp.\ $A^*$). For $0 \leq i \leq d$ define a linear transformation $E_i : V \to V$ such that $E_i v_i=v_i$ and $E_i v_j =0$ if $j \neq i$ $(0 \leq j \leq d)$. Define $E^*_i : V \to V$ in a similar way. The sequence $Φ=(A, \{E_i\}_{i=0}^d, A^*, \{E^*_i\}_{i=0}^d)$ is called a Leonard system on $V$ with diameter $d$. With respect to the basis $\{v_i\}_{i=0}^d$, let $\{þ_i\}_{i=0}^d$ (resp.\ $\{a^*_i\}_{i=0}^d$) be the diagonal entries of the matrix representing $A$ (resp.\ $A^*$). With respect to the basis $\{v^*_i\}_{i=0}^d$, let $\{θ^*_i\}_{i=0}^d$ (resp.\ $\{a_i\}_{i=0}^d$) be the diagonal entries of the matrix representing $A^*$ (resp.\ $A$). It is known that $\{θ_i\}_{i=0}^d$ (resp. $\{þ^*_i\}_{i=0}^d$) are mutually distinct, and the expressions $(θ_{i-1}-θ_{i+2})/(θ_i-θ_{i+1})$, $(θ^*_{i-1}-θ^*_{i+2})/(θ^*_i - θ^*_{i+1})$ are equal and independent of $i$ for $1 \leq i \leq d-2$. Write this common value as $β+ 1$. In the present paper we consider the "end-entries" $θ_0$, $θ_d$, $θ^*_0$, $θ^*_d$, $a_0$, $a_d$, $a^*_0$, $a^*_d$. We prove that a Leonard system with diameter $d$ is determined up to isomorphism by its end-entries and $β$ if and only if either (i) $β\neq \pm 2$ and $q^{d-1} \neq -1$, where $β=q+q^{-1}$, or (ii) $β= \pm 2$ and $\text{Char}(\mathbb{F}) \neq 2$.

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