The end-parameters of a Leonard pair

Fix an algebraically closed field FF and an integer d≥3d≥3. Let V   be a vector space over FF with dimension d+1d+1. A Leonard pair on V   is a pair of diagonalizable linear transformations A:V→VA:V→V and A⁎:V→VA⁎:V→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 ({θi}i=0d,{θi⁎}i=0d,{φi}i=1d,{ϕi}i=1d), called its parameter array. The scalars {θi}i=0d (resp. {θi⁎}i=0d) are mutually distinct, and the expressions (θi−2−θi+1)/(θi−1−θi)(θi−2−θi+1)/(θi−1−θi), (θi−2⁎−θi+1⁎)/(θi−1⁎−θi⁎) are equal and independent of i   for 2≤i≤d−12≤i≤d−1. Write this common value as β+1β+1. In the present paper, we consider the “end-parameters” θ0,θd,θ0⁎,θd⁎,φ1,φd,ϕ1,ϕd of the parameter array. We show that a Leonard system is determined up to isomorphism by the end-parameters and β. We display a relation between the end-parameters and β  . Using this relation, we show that there are up to isomorphism at most ⌊(d−1)/2⌋⌊(d−1)/2⌋ Leonard systems that have specified end-parameters. The upper bound ⌊(d−1)/2⌋⌊(d−1)/2⌋ is best possible.