The end-parameters of a Leonard pair

Fix an algebraically closed field $\text{F}$ and an integer $d\ge 3$. Let $V$ be a vector space over $\text{F}$ with dimension $d+1$. A Leonard pair on $V$ is a pair of diagonalizable linear transformations $A:V\to V$ and $A^{\ast }:V\to V$, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. There is an object related to a Leonard pair called a Leonard system. It is known that a Leonard system is determined up to isomorphism by a sequence of scalars $({\th_i}{i=0}^d, {\th^*i}{i=0}^d, {\vphi_i}{i=1}^d, {\phi_i}{i=1}^d)$,calleditsparameterarray\mathrm{.}Thescalars${\th_i}{i=0}^d(resp.\{\th^i}{i=0}^d$) are mutually distinct, and the expressions $({\mathrm{th}}_{i-2}-{\mathrm{th}}_{i+1})\mathrm{/}({\mathrm{th}}_{i-1}-{\mathrm{th}}_i)$, $(\th^{i-2} - \th^*{i+1})/(\th^_{i-1}-\th^_{i})$areequalandindependentof$i$for$2 \leq i \leq d-1$\mathrm{.}Writethiscommonvalueas$\beta+1$. In the present paper, we consider the “end-parameters” ${\mathrm{th}}_0$, ${\mathrm{th}}_d$, ${\mathrm{th}}_0^{\ast }$, ${\mathrm{th}}_d^{\ast }$, ${\text{vphi}}_1$, ${\text{vphi}}_d$, ${\varphi }_1$, ${\varphi }_d$ of the parameter array. We show that a Leonard system is determined up to isomorphism by the end-parameters and $\beta$. We display a relation between the end-parameters and $\beta$. Using this relation, we show that there are up to inverse at most $\lfloor (d-1)/2 \rfloor$Leonardsystemsthathavespecifiedend-parameters\mathrm{.}Theupperbound$\lfloor (d-1)/2 \rfloor$ is best possible.