Leonard pairs having specified end-entries

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 an ordered 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. Let {vi}i=0d (resp. {vi⁎}i=0d) be such an eigenbasis for A   (resp. A⁎A⁎). For 0≤i≤d0≤i≤d define a linear transformation Ei:V→VEi:V→V such that Eivi=viEivi=vi and Eivj=0Eivj=0 if j≠ij≠i(0≤j≤d)(0≤j≤d). Define Ei⁎:V→V in a similar way. The sequence Φ=(A,{Ei}i=0d,A⁎,{Ei⁎}i=0d) is called a Leonard system on V with diameter d  . With respect to the basis {vi}i=0d, let {θi}i=0d (resp. {ai⁎}i=0d) be the diagonal entries of the matrix representing A   (resp. A⁎A⁎). With respect to the basis {vi⁎}i=0d, let {θi⁎}i=0d (resp. {ai}i=0d) be the diagonal entries of the matrix representing A⁎A⁎ (resp. A  ). It is known that {θi}i=0d (resp. {θi⁎}i=0d) are mutually distinct, and the expressions (θi−1−θi+2)/(θi−θi+1)(θi−1−θi+2)/(θi−θi+1), (θi−1⁎−θi+2⁎)/(θi⁎−θi+1⁎) are equal and independent of i   for 1≤i≤d−21≤i≤d−2. Write this common value as β+1β+1. In the present paper we consider the “end-entries” θ0θ0, θdθd, θ0⁎, θd⁎, a0a0, adad, a0⁎, ad⁎. 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≠±2β≠±2 and qd−1≠−1qd−1≠−1, where β=q+q−1β=q+q−1, or (ii) β=±2β=±2 and Char(F)≠2Char(F)≠2.